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f^-n.-mm.'-Z^: 


I 


PRACTICAL    ESSAYS 


OH 


MILL       WORK 


AND  OTHER 


MACHINERY. 


Q.  WOODFALL  Am9  SON*  amqkl  coubT.  bkimnbk  vtkbbt,  lokooii. 


PRACTICAL    ESSAYS 


ON 


MILL        WORK 


AND  OTHER 


MACHINERY. 


BY   ROBERTSON  BUCHANAN,   Engineer. 


WITH 

NOTES  AND  ADDITIONAL  ARTICLES, 

CONTAININO   NEW    BBSEARCHE8   ON    VARIOUS   MECHANICAL  SUBJECTS^ 

BY  THOMAS  TREDGOLD,  C.E., 

MBMBBB    OP    THE     INSTITUTION    OF    CIVIL    EN0INESB8. 

AND    NOW 

REVISED  INTO  A  THIRD  EDITION  WITH  ADDITIONS, 

BY  GEORGE  RENNIE,  ESQ.  C.E.  F.R.S. 

ETC. 


1- 


lUiU8TRATED   BY   UPWARDS   OF   SEVENTY    PLATES, 

AND    NUMEROUS    FIGURES. 


A'  /-•  \\Z\0 ^ V^         LONDON : 

^      JOHN    WEALE, 

ABCHITECTURAL  LIBRARY,  59,  HIGH  HOLBORN. 

1841. 


publisher's  address. 

lumeH  S  and  4  ;  1750  copies  of  the  two  editions  of  Comte 
dc5  Panibour's  Practical  Treatise  on  Locomotive  Engines ; 
870  copies  of  the  work  on  Bridges  ;  500  copies,  in  a  few 
months,  of  the  work  of  Mr.  Clegg,  Jan.,  on  Coal  Gaa ; 
and  within  one  month  upwards  of  500  of  Mr.  Wicksteed's 
Experiments  on  the  Cornish  Engine  were  sold,  which,  with 
niimy  others,  are  testimonies  of  the  esteem  in  which  such 
works  are  held  at  the  present  time. 

In  the  present  instance  it  affords  me  much  pleasure 
gratefully  to  acknowledge  and  publicly  state  the  liberality 
of  Mr.  George  Rennie,  the  editor  of  this  work,  who,  al- 
though having  multitudinous  professional  engagements, 
has  (anient  in  the  love  of  his  art)  found  the  necessary 
time  for  the  arrangement,  the  addition  to,  and  editing  of 
this  new  edition.  This  has  been  done  gratuitously, 
and  it  is  hoped  that  the  Subscribers,  in  receiving  the  work 
iH^nstHiuently  so  much  cheaper,  will,  in  the  acknowledg- 
ment tvf  its  utility,  respond  to  the  Publisher's  thanks  now 
oxpn^ssoil  for  the  kindness  conferred. 

IkttmKfr  1.1841.  JOHN  WEALE. 


i 


PUBLISHER'S  ADDRESS. 


The  production  of  Works  specially  devoted  to  Engineering 
is,  in  this  country,  frequently  attended  with  difficulty,  not 
arising  from  the  scarcity  of  subject-matter  or  the  disin- 
clination of  practical  men  to  facilitate  its  arrangement,  but 
from  their  inability  to  find  time  to  render  their  willing 
aid.  Delay,  as  in  the  instance  of  the  present  work,  is  in 
consequence  unavoidable.  A  publisher's  risk  is  increased 
in  no  trifling  degree,  when  he  ventures  upon  publications  of 
a  scientific  character  unaided  by  an  author  or  editor  of  ex- 
perience in  the  matters  of  which  they  treat ;  but  it  is  his 
duty  to  select  such  useful  and  novel  subjects  as  shall  not 
only  be  of  practical  help  to  the  engineer,  but  afibrd  a  clear 
view  of  elementary  principles  to  the  student ;  and  as  there 
are  now  amateurs  in  Engineering  as  weU  as  in  other  de- 
partments of  art,  such  works  are  peculiarly  acceptable. 

Keeping  this  twofold  object  steadily  before  me,  it  has  been 
my  oanstant  endeavour  for  many  years  to  render  works  of 
practical  reference  as  complete  as  possible,  especially  by  an 
adequate  number  of  engraved  illustrations  of  examples ; 
and  it  is  with  grateful  feelings  that  I  acknowledge  the 
libenility  of  many  gentlemen,  whose  names  appear  as  con- 
trflmtOFB  of  drawings  in  my  numerous  published  works, 
loaie  few  of  which  are  here  mentioned,  together  with  an 
aoooimt  ct  their  sales :  viz. — ^Tredgold  on  the  Steam  En- 
gDDe,  2S00  copies  since  October,  1838 ;  Public  Works  of 
Great  Britain,  97^  copies  within  the  same  period ;  Papers 
if  tibe  Royal  EngineerB,  1000  copies  of  each  of  the  vo- 


VI  GENERAL    PREFACE. 

corrected  the  Essay  on  the  Teeth  of  Wheels,  and  supplied 
some  additional  tahles  and  a  second  appendix. 

With  a  view  to  practical  utility,  I  have  endeavoured  to 
adapt  the  style  of  these  Essays  to  the  comprehension  of 
such  operative  mechanics  as  have  not  had  the  advantage 
of  mathematical  instruction ;  but  at  the  same  time  I  have 
given  reference  to  authors  for  the  demonstrations  of  such 
propositions  as  I  found  it  necessary  to  introduce,  in  order 
to  give  such  workmen  some  notion  of  the  principles  on 
which  their  work  should  be  conducted. 

For  any  repetitions,  want  of  unity,  and  other  imperfec- 
tions, which  will  doubtless  too  readily  appear  in  these 
Essays,  I  may  offer  the  same  apology  which  I  did  on  a 
former  occasion,  that  they  were  written  at  many  Afferent 
and  distant  intervals,  occasioned  by  intemipticms  from 
professional  and  other  engagements. 


Note. — The  Second  Edition  was  superintended  by  the  hite  Mr.  Tred- 
gold ;  and  the  principal  facts  noticed  in  it  are  incorporated  in  the  following 
Preface. 


PREFACE 


TO  THB 


THIRD    EDITION. 


The  Essays  of  Robertson  Buchanan  on  Practical  Me- 
chanics have  been  long  known,  and  duly  appreciated  by 
the  public 

They  consist  of  a  series  of  treatises,  seven  in  number, 
on  several  of  the  elementary  parts  of  machinery ;  such  as 
the  Teeth  of  Wheels,  published  in  1808 ;  on  the  Shafts  of 
Mflls,  in  1809 ;  and  on  Millwork  and  other  Machinery,  in 
1814.  The  copious  Index  of  the  contents  of  the  present 
edition,  drawn  up  by  Dr.  Jamieson,  sufficiently  explains 
the  nature  of  the  work. 

In  perusing  the  Essay  on  the  Configuration  of  the  Teeth 
of  Wheels,  we  are  at  once  struck  with  its  resemblance  to 
the  admirable  treatise  of  Camus,  published  in  1782%  and 
which  the  author  duly  acknowledges.  The  subject  is 
divided  into  two  parts ;  firstly,  the  principles  as  laid  down 
by  Camus ;  secondly,  the  application  of  these  principles  to 
difEsreiit  kinds  of  spur  and  bevel  gear. 

*  Coon  de  Math^matique. 


TUl  PREFACE   TO   THE 

The  first  application  of  the  epicvcloidal  curve  to  the 
teeth  of  wheels  is  generally  ascrihed  to  Roemer,  a  Danisii 
mathematician,  in  1674^%  although  De  la  Hire^  claimed  the 
merit,  and  demonstrated  that  if  a  tooth  of  either  a  wheel 
or  pinion  be  formed  by  a  portion  of  an  exterior  epicycloid, 
described  by  a  generating  circle  of  any  dimensions  what- 
ever, the  tooth  of  its  follower  will  be  properly  formed  by  a 
portion  of  an  interior  epicycloid,  described  by  the  same 
generating  circle.  The  object  he  had  in  view,  was  to  se- 
cure a  perfect  uniformity  of  pressure  and  velocity  to  the 
machine,  so  that,  in  all  positions,  the  wheels  which  trans- 
mit the  power  should  act  equally  and  similarly,  and  (hat 
the  surfaces  of  the  teeth,  by  touching  in  a  point,  should 
roll  over  each  other  when  in  motion,  and  thus  avoid  all 
friction,  a  desideratum  hitherto  impracticable  to  ac- 
complish. 

The  general  properties  of  the  cycloid  and  epicycloid, 
and  the  modes  of  generating  these  curves,  both  geometric- 
ally and  mechanically,  have  been  given  by  various  authors, 
and  Buchanan  applied  the  principles  of  Camus  to  the  forms 
of  the  surfaces  of  the  teeth  of  wheels  and  pinions  acting 
against  each  other  imder  different  circumstances ;  such  as 
the  wheel  and  trundle,  the  wheel  and  pinion,  the  rack  and 
pinion,  conductor  or  conducted,  external  or  internal,  spur 
or  bevel  gear,  so  as  to  render  them  comprehensible  by  the 
general  reader.  The  author  next  investigates  the  action 
of  conical,  or  bevel  wheels,  under  the  different  circum- 
stances of  the  inclination  of  their  axles,  applying  the  same 
exterior  forms  of  the  teeth  of  spur  wheels  to  the  teeth  of 
bevel  wheels,  with  the  exception  that  the  curves  should  be 
a  spherical  epicycloid.  The  principles  of  bevel  wheels  had 
lieen  already  pointed  out  by  De  la  Hire  in  the  year  1666  % 

*  Wolfii  Opera  Mathcmaticis. 

^  Traite  des  Epicycloidcs,  1694. 

e  Mcmoires  de  rAcadcmie,  1666.  1669. 


THIRD    EDITION.  ix 

igh    long   knonn   and    applied    previously*,    in    the 
p  of  a  conical  tnmdlc. 

De  la  Hire''  was  not  only  among  the  first  to  apply  the 
«?picycloid  to  the  configuration  of  the  teeth  of  wheels,  but 
le  considered  the  involute  of  a  circle,  as  the  best  of  the 
«xterior  epicycloids,  and  which  it  may  be  proved  to  be, 
if  we  consider  the  generating  straight  line  as  a  curve  of 
I^H  infinite  radins,  and  which  would  strictly  apply  to  a  pinion 
^Baeting  on  a  rack,  and  vice  versa  to  the  teeth  of  a  rack. 
^HEoler,  in  I76O,  treated  the  case  of  the  involute  very 
^Bgcoerally '. 

^B  Kaestner*,  in  1771.  shewed  a  method  of  describing 
1^^  and  applying  the  involute  to  the  teetli  of  wheels.  Pro- 
fessor Kobison  applied  the  involute  to  the  wheels  of  a 
miH  near  Edinburgh,  but  the  result  was  any  thing  but 
■Satisfactory.  The  same  principle  has  since  been  advo- 
by  Ferguson ",  Professor  Airey ',  and  Professor 
I        WiUis,  whose  valuable  paper  appears  in  the  Appendix. 

We  are  also  indebted  to  several  continental  writers,  but 
^■^particularly  to  M.  Hachctte',  for  his  elaborate  investigation 
^Bnf  the  curves  most  applicable  to  the  teeth  of  wheels.     Pro- 
^Bfessor  Airey  states,  "  That  in  order  that  the  mechanical  effect 
which  one  wheel  will  produce  upon  another,  may  in  all 
positions  be  the  same,  it  is  necessary  that  the  line  perpen- 
dicular to  the  surfaces  of  the  teeth  at  the  point  of  contact, 
intersect  the  line  joining  the  centres  at  a  fixed  point,  which 
divides  that  line  into  two  parts,  the  ratio  of  which  is  the  me- 


•  Beswni,  1582.     Tlicatnim  Mafliinnrum. 

*  Traite  Jm  Epicycloids,  and  Novo  ConuneDt.  Petropol.  1754,  1755. 
■  Comment.  Petro|>o].  1T5+,  1755. 

*  De  Denubus  Botorum  Reg.  Soc.  Goltingensis. 

•  Sir  Dnvid  Brewster,  edition  of  1807. 

'  Cambridge  Pliilosojiliicai  TransnctionB,  Vol.  U. 
»  Tniite  felemcntaire  dcs  Machines,  181 1. 


X  PREFACE   TO   THE 

chanical  power ;  when  this  holds,  the  proposition  of  the  an- 
gular velocity  will  be  constant."  Mr.  Airey  then  deni(Hi- 
strates  the  case  mathematically,  and  advises  tiiat  the  teeth 
be  made  to  work  a  litde  before  and  after  tiie  line  of  cen- 
tres, and  thinks  that  a  tooth  formed  by  the  union  of  an 
epicycloid  and  hypocycloid  is  preferable  to  any  form  what- 
ever, for  tiie  line  of  action  is  always  very  nearly  perpen- 
dicular to  the  radius,  by  which  means,  not  only  is  the  fric- 
tion made  much  less,  but  also  the  strain  upon  tiie  axles  is 
considerably  diminished.  The  same  applies  to  bevel  wheels 
and  rack-work,  with  reference  to  uniformity  of  motion  and 
action,  which  he  conceives  to  be  of  far  greater  consequence 
than  any  diminution  of  friction,  which  can  never  be  re- 
duced to  nothing,  except  the  part  of  contact  be  always  in 
the  line  of  centres,  a  condition  which  may  be  satisfied  only 
by  means  of  an  infinite  number  of  curves,  and  amongst 
others  by  two  logarithmic  spirals,  but  the  mechanical  ac- 
tion, and  the  motion  would  be  dreadfully  irr^ular. 

This  question  is  now  littie  more  than  one  of  mere  cu- 
riosity, arising  from  the  smallness  of  the  teeth  of  wheels 
now  made,  and  the  greater  perfection  of  workmanship  in 
the  materials,  in  consequence  of  the  use  of  iron  wheels, 
and  the  accuracy  with  which  the  teeth  are  formed  and  ad- 
justed by  the  most  simple  method  of  templates  and  com- 
passes ;  and  the  approximation  of  the  form  thus  generated 
to  the  form  presented  by  theory  is  very  close.  The  deter- 
mination of  the  strength  of  the  teeth  to  the  power  to  be 
transmitted,  is  given  in  the  fourth  chapter  on  the  principles 
of  proportioning  the  strength  of  the  teeth  of  wheels. 

The  rule  adopted  by  millwrights  for  finding  the  depth  of 
the  teeth  from  the  bottom  to  the  pitch  line,  and  from  the 
pitch  line  to  the  top  of  the  tooth,  is  simply  to  multiply  the 
pitch  by  5,  and  divide  by  9,  and  vice  versd. 

The  curves  of  the  exterior  and  interior  surfaces  of  the 


THIRD    EDITION.  XI 

a  are  sometimes  traced  by  means  of  a  tracer  fixed  in 
the  radius  line  of  one  of  two  pieces  of  board,  the  edges  of 
IJL  which  arc  cut  out  to  suit  the  primitive  circles  of  the  wheels 
^■required ;  then,  by  fixing  a  template  to  one  edge,  and  di- 
^^riiUng  the  teeth  accurately,  the  tracer  will,  by  the  rolling 
I        of  the  two  circles,  describe  the  curves  required. 

The  usual  mode  of  describing  the  teeth  of  wheels  by 
arcs  of  circles,  is  admitted  to  approximate  to  the  true 
cur\'e,  if  the  centre  and  radius  of  the  wheel  be  deter- 
mined correctly.  In  the  best  establishments,  this  is  invari- 
ably done,  and  the  result  of  many  years'  practice  has  proved 
the  goodness  of  the  system. 

With  respect  to  Buchanan's  tables  on  the  pitches  of 
wheels,  and  the  strenjrth  of  the  teeth  with  the  correspond- 
ing numbers  and  horses'  power  moving  at  the  ])itch  line  at 
different  velocities,  it  will  be  observed  that  the  prevailing 
proportion  is,  that  the  pitch  is  about  double  the  thickness 
of  the  tooth,  and  the  length  rather  longer  than  the  thickness, 
but  three  to  four  times  the  thickness  is  more  usually 
^K^diopted. 

^B     The  investigation  of  the  pr()i)er  curves  to  be  given  to  the 

^■teeth  of  wheels,  by  Professor  Willis,  has  been  added  by  way 

of  an  appendix  to  the  concluding  chapter  on  the  Teeth  of 

Wheels.     The  first  section  gives  a  succinct  account  of  the 

curves  adapted  to  practice,  and  shews,  by  way  of  corollary, 

that  if  for  a  set  of  wheels  of  the  same  pitch  a  constant  de- 

^^  Jcribing  circle  be  taken,  and  employed  to  trace  those  por- 

^Btions  of  the  teeth  which  project  beyond  each  pitch  line  by 

^Vmlling  on  an  exterior  circumference,  and  those  which  lie 

within  it,  by  rolling  on  its  interior  circumference;  then 

any  Iwo  wheels  of  this  set  will  work  correctly  together. 

I  Profesgor  Willis  then  shews  how  this  can  be  accomplished, 

^'ind  then  gives  a  form  of  increased  strength  to  the  backs  of 

tlie  teeth,  but  which  arc  only  suited  to  move  in  one  diroc 

The  second  section  of  this  paj)cr  shews  how  the 


XU  PREFACE   TO   THE 

practical  approximation  to  the  true  form  can  be  accom- 
plished by  arcs  of  circles,  a  form  which  approaches  to  mo- 
dem practice.  An  instrument  termed  an  odontagraph, 
together  with  tables  for  facilitating  its  use,  and  for  forming 
cutters  for  shaping  the  teeth,  is  proposed,  and  a  theory  given 
of  the  nature  of  the  motion  which  is  produced  by  the  pres- 
sure of  one  circular  arc  upon  another,  when  disposed  so  as 
to  work  in  the  manner  of  teeth. 

The  Essay  on  the  Shafts  of  Mills  is  divided  into  five 
chapters,  containing  a  general  description  of  shafts  most 
employed  in  miU-work,  and  the  strains  to  which  they  are 
subject  from  lateral  stress  and  torsion ;  the  strength  and 
stiffening  of  shafts,  journals,  and  gudgeons,  with  refer- 
ence  to  the  strength  of  materials,  according  to  the  ex- 
periments of  different  authors.  The  subject  of  torsion  is 
briefly  examined,  in  conjunction  with  lateral  stress.  It  is 
shewn  that,  in  general,  the  strength  of  a  cylinder  or  solid 
axle  to  resist  the  force  of  torsion,  is  as  the  cube  of  its 
diameter,  and  that  the  length  of  a  cast  iron  shaft  has  no  in- 
fluence on  its  resistance  to  torsion,  whatever  may  be  the 
exception  with  wooden  shafts.  The  power  of  a  cast  iron 
shaft  to  resist  torsion  is  calculated  firom  Mr.  Tredgold*s 
formula,  which  considers  the  resistance  the  same  as  firom 
the  lateral  stress.  The  Table  of  Shafts,  at  the  end  of  the 
fifth  Chapter,  takes  into  consideration  the  two  kinds  of 
resistance. 

The  use  of  iron  in  machinery  previously  to  its  adoption 
in  England,  is  evidenced  by  referring  to  the  works  of 
RameUi,  Bockler,  and  Bessoni,  where  there  are  repre- 
sentations of  iron  wheels,  and  portable  miUs  and  cranks 
of  the  same  metal,  but  it  was  only  used  in  this  country 
about  the  \^^ar  1550. 

Iron  pipotsi  canio  into  use  in  France  about  the  year  I672, 
aiul  fKun  the  UHur  178^2  to  17S4,  cast  iron  was  used  in 
machinery  at  Culobrook  Dalt\  Rotherfaam,  and  at  most  of 


THIRD    EDITION.  "XIU 

e  gtetA  iron  works  in  England.     Cast  iron  wheels  were 
silso  in  iise  at  Manchester,  Liverpool,  Nethcrhy,  and  several 

k  other  places. 
The  theory  of  torsion  has  been  investigated  by  several 
writers,  but  with  very  little  effect.     Coulomb  was  the  first 
lo  direct  the  attention  of  mathematicians  to  this  kind  of 
Stsistance.     If  a  cylindrical  body,  such  as  a  line  or  series 
i,        of  lines  or  fibres  be  suspended  vertically,  but  having  its 
upper  end  fixed,  be  turned  round  through  any  angle  by 
1^^  the   existence  of  some  lateral  force,  and  if  its   elasticity 
^B<bc  not  impaired,  it  will,  after  the  deranging  influence  has 
^Vccftsed,  return  to  its  former  position,  and  may  perform  this 
retrocession  in  a  certain  time.     Professor  Leslie ',  in  con- 
sidering this  question,  conceived  the  cylinder  to  consist  of 
^_  a  series  of  thin  discs,  which,  when  twisted,  each  successive 
^^^sc  will  make  an  angular  advance  till  this  accumulation 
^Vst  a  certain  distance  amounts  to  a  complete  circuit,  and 
'T       such  revolutions  will  be  repeated  at  every  like  interval 
The  torsion  is  thus  proportionate  to  the  angle  of  deviation 
and  the  fourth  power  of  the  diameter  of  the  cylinder,  and 
inversely  as  its  height. 

If  the  exterior  particles  of  a  line  or  series  of  lines  be 

lOverstrained,  then  their  cohesion  will  be  destroyed,  and  the 

ine  return  no  longer  to  its  pristine  state.     If  the  strain  be 

rithin  the  limits  of  the  elasticity  of  the  exterior  particles, 

[the  oscillations  will  be  perfectly  isochronous. 

The  application  of  this  principle  by  Coulomb''  led  to  the 
istmction  of  an  exquisite  balance  for  detecting  and 
tneasuring  the  smallest  forces.  Thus,  by  establishing  a 
connexion  between  the  elastic  and  passive  resistance  of 
metals,  and  the  resistance  &om  torsion,  we  are  led  to  an 
approximation  to  the  true  theory.      The  experiments  of 

•  Eienieots  of  Nntuml  Pliilosophy. 

Meinoircs  de  TAcademJc  dcs  Sciences,  178*. 


•XIV  PREFACE   TO    THE 

Minard*  and  D^sormes,  Lagerhjelm^,  Bomet%  Segum^  and 
Ardant%  however,  effected  much  towards  the  solution  of 
the  question. 

The  experiments  of  Savart*  on  the  torsion  of  different 
bars  show  the  following  results : 

1st.  That  the  angles  of  torsion  are  in  every  case  pro- 
portionate to  the  forces  of  torsion  within  the  limits  of 
elasticity. 

2dly.  That  in  bars  of  the  same  section,  the  forces  of  tor- 
sion are  directly  proportionate  to  the  length  of  the  bars, 
the  length  having  little  other  influence  than  increasing  the 
angle  of  torsion. 

The  experiments  by  Duleau,  Banks,  Dunlop,  and  by 
the  Editor  of  this  work,  approximate  pretty  nearly  the 
mean  values. 

On  the  dynamical  effect  of  men  and  horses,  Buchanan 
quotes  the  estimates  of  Desaguliers,  Emerson,  Smeaton, 
and  Watt,  as  also  a  table  in  units  of  force  from  Dr.  Young. 
Buchanan's  own  experiments  of  the  effective  power  of  men 
working  in  different  positions,  had  already  been  published 
in  the  fifteenth  volume  of  the  Repertory  of  Arts,  for 
1801.  Of  the  authors  who  have  written  upon  this  sub- 
ject, the  names  De  la  Hire ',  Amontons ',  Lambert  \  Ber- 
noulli ^  Coulomb  \  and  Schulze,  are  most  distinguished. 
Do  la  Hire  considered  the  question  almost  entirely  with  re- 

*  Annales  do  Chimie. 

^  Du  Fer  dans  lee  Fonts  suspendus. 

^  Poncelet,  Mecanique  Industrielle,  1839. 

^  Des  Fonts  en  Fils  de  Fer,  1826. 

*  Annales  de  Chimie. 

'  M^moires  de  TAcadeniie  des  Sciences,  1699. 
K  M^moires  de  I'Academie  des  Sciences,  1709. 
^  Mcmoires  de  TAcademie  de  Berlin,  1776. 

*  Mcmoires  de  TAcademie  des  Sciences,  1760. 
^  Prix  do  TAcademie,  torn.  vii.  1753. 


THIRD    EDITION.  •jy 

(  to  the  muscular  powers  of  animals.  Amontons 
with  reference  to  their  velocity.  Deparcicux  their  weight 
and  muscular  action  conjoiutly,  and  Lambert  reduced  the 
problem  to  an  equation,  which  gives  the  relation  between 
the  weight,  velocity,  burthen,  and  path  of  inclination 
pursued  by  the  animal.  Bernoulli  maintained  that  the 
quantitj-  of  fatigue  was  always  proportional  to  the  quan- 
tity of  action,  whatever  be  the  nature  of  the  work  per- 
fonoed. 
Coulomb  proved  the  absurdity  of  this  position,  by  shew- 

fing  that  the  daily  quantity  of  action  is  variable  according 
to  the  circumstances  under  which  it  is  developed,  and  by  a 
iKries  of  valuable  experiments,  comprehending  almost  everj- 
ease,  be  assigned  values  to  each  kind  of  action. 
Schulzc's  investigations  on  the  absolute  and  mean  effect 
of  human  labour  to  lift,  draw,  and  push  weights,  have  fur- 
oished  us  with   standards  of  value  which  may  be  safely 
■  taken  as  the  average  of  continental  labour. 
^H    The  more  modem  investigations  of  Gerstner  \  Morin, 
^  Tourier,  Devilliers,  and  others  on  the  continent,   and  of 
Trcdgold,  Palmer,  Sylvester,  Bevan,  M'Neill,  and  Field,  in 
this  countrj",  besides  many  experiments  which  have  been 
made  by  ourselves  ^  on  the  extreme  and  mean  powers  of  men 
and  horses,  have  established  sufficient  data  whereon  to  found 

'  Treadse  on  ME^haoics,  tronslBtcd  from  the  Oerm&n,  1834. 
''  TliB  foUowing  ia  a.  brief  statement  of  the  dynamical  effect  of  human 
1  liorec  power  aevcrelly  dppUed  to  wulki&g  whcel-croncs,  cruba,  cranes, 
FpI^^Tifig  engines,  hor»e  runs,  &c.,  for  raising  iliffcrent  materials. 

MANUAL  LABOUR. 

BuiLDiNo  Matkrials. 

I  iHlj.  One  man  in  67  juunicys  raised  a  weight  of  16,342  lbs. 

(incluiUtig  Uie  weight  of  liiniiSclF]  to  tlic  height  of  30  feot  in  Ibf). 

lOhonis,  oiual  lo  a  weight  of 817 

niW  I  fuot  high  jwr  niinute. 


•Xvi  PREFACE   TO   THE 

tiprmaht  suitable  to  <fverv  cage  of  the  apfdicatioD  of  animal 
|Mnri$r  Up  perfirirm  mechanical  operaticMi& 

S^J/.  Ocme;  nuMi  in  47  journepi  raised  a  wei^  of  11,374  lbs. 

(imludUig  bk  own  weij^t)  to  a  hei^  of  50  feet  in  10  honn,  lbs. 

Mltml  Uf  a  wei^t  of 974.8 

nimi  1  foot  ttijg^  per  minate. 

OKniNAfty  Ciuifis. — Expsbimbnts  madb  at  thb  Wbbt  India  Docks. 

'Idly.  Tlio  power  of  six  men  applied  to  a  crane  is  capable  of        ^ 
rakitig  224,000  Ibt.  15  feet  high  in  eight  hours,  equal  to  a 

witiglitof 1166.6 

rained  1  foot  high  per  minute  by  1  man. 

4thly.  Tlio  power  of  0  men  applied  to  a  crane  is  capable  of 
nuNJiig  a  weight  of  262,080  lbs.  12  feet  high  in  8  hours,  equal 

to  a  weight  of 1092 

raiitefi  1  foot  high  |>er  minute  by  1  man. 

N,li*  Hy  experiment,  the  friction  of  these  cranes  varied  from 
^Qth  to  j^^tli  of  tlio  absolute  weight. 

OhDINAHY  ChANKS.— ExPBRIMBNTS  IfADB  AT  THB  LONDON   D0CK8. 

Athly.  Hy  tlio  ^idking  whool-crane,  worked   by  6   men,   a 

wolglit  of  787»920  lbs.  >i-as  raised  7  feet  in  8  hours,  equal  to  1915 

ndmfiA  1  foot  high  per  minute  by  1  man. 
(tthly.  Again,  by  tlio  >i*alking-wheel  crane,  worked  also  by  6 

mon,  a  >\*t>ight  of  91  K680lb6.  was  raised  8  feet  in  11  hours, 

(M)ual  to 1841 

iniMHl  \  foot  high  por  minute  by  1  man. 
?thly«  Hy  t  orabu  >i^Mrkcd  by  6  men  each,  a  ^-cight  of  728,000  lbs. 

WHN  miw^l  to  a  height  of  16  fe«l  in  8  hours*  equal  to    .  2012 

nuii«Hl  \  (\H\t  higli  |^>r  mimiti!'  by  1  man. 

HORSE  POWER. 

SihK\  TW  vKiMttiknJ  cdixt  of  a  horse  power  appBed  to  a  pile- 
\lri\u^  o^ue  >{k\vrk«^  by  i  iMneiv  was  fNuad  to  be  equal  to 
M  \H\H$bt  vxt'  4:(^A^  cKjk  vaib«ol  1  jRwt  bi^  ui  ;^  MceaAi^  ar  a 
>m\^^t^a' »AS9 


THIRD    EDITION.  *XVU 

The  Appendix  to  the  Second  Essay  contains  tables,  by 
iTrcdgold,  on  the  properties  of  materials,  and  the  influence 
f  alloys  in  increasinj!;  the  tenacity  of  metals.  These  data 
lave  been  farther  extended  by  the  experiments  of  Messrs. 
tairbaim  and  Hodgkinson  on  the  relative  strength  of  hot 
od  cold  blast  iron  %  and  on  the  compression  of  cast  iron 
nlumns*.  But  our  knowledge  of  the  elastic  properties,  of 
materials,  the  laws  of  the  elongation  and  compression, 
and  the  effect  of  temperature  upon  their  cohesion,  is  as  yet 
hut  imperfectly  known.  The  experiments  of  Rondelet, 
Dupiu,  Tredgold,  Barlow,  Bramah,  Gerstner,  and  Adam 
Burg  on  the  flexure  and  resilience  of  wood  and  iron  have 
Tiishcd  some  valuable  facts  on  this  subject,  but  it  is  to 
Messrs.  Minard,  Desormes,  and  Ardant  that  we  are  prin- 
Uy  indebted,  for  determining  the  law  of  elongation  by 
direct  tension  of  the  fibres  of  wood  and  iron.  Vicat 
lowed,  in  the  case  of  a  cubical  prism  of  lead,  that  the  law 
r  compression  ie  constant  from  a  constant  augmentation 
f  pressure.  Peclet  proved  that  for  c^ast  iron  the  molecular 
Uplaccment  of  the  crystals  did  not,  in  the  first  instant 
r  compression,  exactly  follow  the  compression  in  propor- 
to  the  resistance,  and  our  own  experiments  in  the 
compression  of  several  of  the  softer  metals  have  shewn  the 

Btliljr.  Again,  the  power  of  a  horse  applied  to  working  runs  for 

fcninug  earthwork  up  a  riui  or  inclined  plane,  the  hone  of 
nliidi  was  60  feet,  and  the  vertical  height  10  feet,  was  equal 
to  a  resistance  of  ilOlhs.  trnvelling  through  a  space  of  72  feet  lbs. 

ID  I  nunatc  by  two  horses,  which  ia  equal  to        .         .         .        14,760 
ntted  1  fool  high  per  horse  power  per  minute ;  a  result  very 
inferior  to  the  laat,  arising  from  the  inconstant  nature  of  the 

»  S«rtnth  Report  of  the   British  Association  for  the  Adi 

k  Bxpcrimental  Researches  on  the  Strength  of  Ptllora  of  Cast  1) 
dF  otber  MctcriaJd.— Philosophical  Transacdonii,  1840. 


•xviii  PREFACE   TO   THE 

densities  to  have  increased  in  a  greater  ratio  than  the  com- 
pression. 

The  influence  of  temperature,  so  far  as  the  temperature 
of  the  atmosphere  is  concerned,  appears  to  exercise  very 
little  influence,  but  when  carried  beyond  the  limit  of  at- 
mospherical temperature,  the  experiments  of  Messrs.  Tre- 
mery  and  Poirier  have  shewn  that,  at  a  dull  red  heat,  (450*" 
Fahrenheit,)  the  tenacity  of  a  bar  of  iron  had  lost  one 
sixth  of  its  original  strength. 

M.  Savart  *  proved,  by  means  of  a  series  of  ingenious  ex- 
periments on  the  sonorous  vibrations  of  difierent  materials^ 
the  influence  of  time  in  the  aggregation  of  the  particles  in 
cooling  of  substances,  apparently  homogeneous ;  and  Messrs. 
Vicat,  Minard,  and  Desormes,  and  ourselves  ^  have,  by 
means  of  iron  bars  loaded  to  within  the  limits  of  their  ab- 
solute strength,  shewn  that  permanent  set  or  loss  of  elasti- 
city, and  even  rupture,  takes  place  when  influenced  by  time. 

On  the  subject  of  Shafts  and  Couplings,  a  new  era  had 
arisen.  The  introduction  of  the  textile  fabrics  in  the 
country,  by  Lombe  and  others,  and  the  inventions  of  Wyatt» 
Arkwright,  and  Watt,  led  to  a  new  system  of  machinery. 
The  necessity  of  producing  high  velocities  occasioned  a  cor- 
responding diminution  in  the  dimensions  of  shafts,  and 
those  ponderous  masses  of  wood,  cast  iron,  and  their  enor- 
mous bearings  and  couplings,  gave  place  to  slender  rods  of 
wrought  iron  and  light  frames  or  hooks  for  suspending 
them.  In  like  manner,  wheels  and  pulleys  of  large  dia- 
meters were  replaced  by  pulleys  and  straps  of  moderate  dia- 
meters and  dimensions,  and  by  uniting  the  pulleys  in  series 
of  difierent  diameters,  and  alternating  their  positions  oppo- 

*  Annales  de  Ghimie  et  de  Physique,  snr  les  Vibrations  longitadinales  des 
Corps,  tome  65. 

^  On  the  Effects  of  Temperature  on  the  Arches  of  Bridges.  Tnmsaotiinns 
of  the  Institntion  of  Civil  Engineers,  Vol.  III. 


THIRD    EDITION-.  *XIX 

I  each  other,  a  greater  variety  of  velocities  were  ob- 
ncd,  and  a  great  deal  ttf  friction  and  noise  done  away  with, 
rithout  taking  into  conaideratioii  the  economy  resulting 
IVoin  the  lighter  kind  of  machincrj'  and  the  less  quantity 
of  power  than  formerly  required  to  put  the  wliole  in  motion. 
"besB  improvements  are  in  a  great  measure  due  to 
.  Fairhaim  and  Lillie".  To  use  the  words  of  Dr. 
"  The  method  of  increased  velocities  in  the  driving 
i  of  factories  is  undoubtedly  one  of  the  most  remarkable 
nprovements  in  practical  dynamics.  It  diminishes  greatly 
the  inertia  of  the  mass  to  be  moved,  by  giving  to  much 
lighter  shafts  and  wheels  the  same  momentum,  and  it  per- 
raitii  the  pulleys  or  drums  which  immediately  impel  the 
machines  by  straps*  to  be  reduced  to  a  size  much  nearer  to 
lat  of  the  steam  pulleys  fixed  on  the  main  axes  of  these 


I  The  same  improvements  have  taken  place  with  regard 
B  the  couplings,  which  are  now  reduced  to  simple  rings 
f  wrought  metal  keyed  to  the  circular  ends  of  the  abutting 
ids  of  the  shaiU. 

•  In  a  letter  to  the  Editor  of  this  pnblicatioo,  Mr.  F&irbaJm  dates  the 

action  of  llie  new  system  of  gearing  from  the  year  1815:  at  that 

B,  wyn  he.  "  the  ahufts  of  our  cotton  mills  were  moving  at  40  and  50  re- 

is  per  minute,  whereas  at  the  present  day  we  hove  none  under  60, 

u  numy  as  300  and  350.     The  same  number  of  revolntionK  are  appli- 

i  now  in   use  for  flnl  oud  Bilk.      The  extensive  use  of  wrought 

Kbon  for  (bafts,  and  the  slide  lathe,  bnve  ^ven  wonderful  facilities  to  the 

r  p*iurtioii  of  ihofts,  and  increased  velocities  and  reduced  friction  by  the 

•Bwrniwoti  of  great  power  through  a  comparatively  small  section.     In 

nnr  rif  the  more  recent  mills  of  iny  coiistrucdon,  we  have  shafts  only  2^ 

r  overconiing  the  power  of  a  iO-horse  engine.     Another 

aDpnivinicnt  n-iu'  our  system   of  coupling,  and  the    mode  of  suspending 

I   A*Ai  fmn  tliG  main  beams  and  ceilings  of  rooms,  &c.     In  the  first  instance 

f  never  get  loose,  and  In  tlie  second,  the  shafts  ai'e  strung  like 

'   *)ns  ibittg  the  celling,  and  with  small  iron  pulleys  transmit  the  motion  to 

I  At  udiincry  without  crowding  the  room  or  obstructing  the  light." 

*  PUlMOphy  nf  Manufactures. 


Aft 


•XX  PREFACE   TO   THE 

As  respects  iron,  cast  iron  pipes  and  cranks  and 
pumps  were  used  in  the  old  London  Bridge  Water- 
works, by  Sorocold,  in  the  time  of  Charles  II.,  and 
mention  is  made  of  a  cast  iron  wheel,  4  feet  in  diameter, 
which  worked  into  a  pinion  6  inches  in  diameter ;  and  he 
adds,  **  If  the  teeth  of  the  wheel  be  of  brass,  and  the  teeth 
of  the  leaves  of  the  pinions  of  iron,  the  machine  will  work 
more  equally."  It  seems  generally  believed,  however,  that 
Smeaton  was  the  first  to  introduce  cast  iron  wheel  work  in 
machinery  at  the  Carron  Iron  Works,  for  the  purpose  of 
boring  cannon,  about  the  year  1769>  although  he  had  pre- 
viously applied  a  cast  iron  axis  for  a  windmill  in  17^4 ; 
but  the  founder's  art  was  so  imperfect,  that  Smeaton  was 
obliged  to  proceed  cautiously:  and  it  was  not  until  the 
years  1784  and  1785,  when  the  Albion  Mills  were  built, 
that  cast  iron  was  applied  to  all  parts  of  machinery,  and 
the  late  Mr.  Rennie  was  the  first  to  introduce  accuracy  in 
the  forms  of  the  teeth  of  wheels,  by  turning  and  adjusting 
the  teeth,  and  causing  the  iron  to  work  into  wooden  cogs. 
The  subsequent  progress  which  has  been  made  in  the 
later  period  of  his  life,  introduced  a  new  era  in  mill  ma- 
chinery, which,  in  point  of  accuracy  and  smoothness  of 
workmanship,  has  not  been  exceeded,  even  under  the  au- 
tomatic system  of  self-acting  tools.  Arkwright  used  iron 
bevel  wheels  and  band  pulleys,  at  the  cotton  spinning  mills 
at  Cromford  and  Helper,  in  1775. 

The  Fourth  Essay  of  Buchanan,  on  the  Method  of  Disen- 
gaging and  Re-engaging  Machinery  while  in  Motion,  may 
be  fairly  included  in  the  Third  Essay  on  Couplings,  with  the 
exception  of  the  fast  and  loose  pidleys  and  friction  clutches, 
which  are  found  to  be  the  simplest  and  best  for  engaging 
and  disengaging  machinery  without  shocks.  The  friction 
plate  inclosed  between  two  other  plates,  introduced  some 
years  back  by  ourselves,  has  been  found  to  answer  all  the 
conditions  in  point  of  simplicity  and  efiect  required  by  a 


THIRD   EDITION.  •xxi 

friction  pulley,  and  does  away  with  all  the  ineonyeniences 
of  the  cones. 

The  Fifth  Essay  on  the  Mechanism  for  equalizing  the  Mo- 
lion  of  Mills,  relates  to  the  changes  of  velocity  to  which  every 
first  mover  is  suhject,  either  from  an  increase  or  diminution 
in  the  supply  of  power,  or  where  the  power  is  uniform,  from 
the  increase  or  diminution  of  the  resistances  required  to  he 
overcome.  This  is  accomplished  by  means  of  double  or 
conical  pendulums  and  balls,  cither  for  regulating  the  sup- 
ply of  wind,  water,  or  steam,  according  to  the  quantity  of 
action  required. 

The  Appendix  to  the  Fifth  Essay  is  extracted  from  a 
paper  communicated  by  Buchanan,  in  the  year  1799,  to 
the  Philosophical  Society  of  Edinburgh,  and  afterwards  to 
the  Editor  of  the  Philosophical  Magazine,  on  the  Velocity 
of  Water  Wheels.  The  author  negatives  the  conclusions 
of  Banks,  viz.  that  the  velocity  of  an  overshot  wheel  is  as 
the  cube  root  of  the  quantity  of  water  it  receives,  by  con- 
trasdng  his  own  experiments  on  water  wheels  moving  with 
their  common  velocity  and  half  that  velocity ;  and  the  re- 
sult was,  that  the  last  half  required  just  half  the  quantity 
that  the  first  did ;  and  this  he  confirms  bv  two  letters  from 
Mr.  Robcrton,  in  which  the  author  contrasts  the  maximum 
velocities  of  Smeaton  and  Banks's  water  wheel ;  and  says  that 
while  Smeaton,  by  his  maximum  velocity  of  throe  feet,  lost 
only  one-fourth  of  the  original  effect,  Banks,  at  his  maximum 
velocity  of  one  foot  per  second,  reduced  it  to  one  half  of 
that  velocity,  thus  making  the  same  quantity  of  water  pro- 
duce four  times  the  quantity  of  work,  or  twenty  times  the 
quantity  of  work  which  Smeaton  could  perform  with  the 
same  quantity  of  water.  The  continuation  of  Buchanan's 
Appendix  shews  that  the  mechanical  effect  depends  on  the 
wheel's  diameter,  the  height  of  the  fall,  and  on  the  velo- 
city of  the  circumference  of  the  wheel ;  and  it  is  shewn 
that  a  water  wheel  will  produce  the  greatest  effect  when 


•XXU  PREFACE   TO    THE 

the  diameter  of  the  wheel  is  proportioned  to  the  height  of 
the  fall,  so  that  the  water  flows  upon  the  wheel  at  a  point 
about  52f  degrees  distant  from  the  summit  of  the  wheel. 

The  subject  of  water  wheels  has  been  fully  treated,  both 
theoretically  and  practically,  by  many  authors  both  on  the 
continent  and  in  this  country ;  suflSce  it  to  mention  the 
names  of  Pitot,  Deparcieux,  Lambert,  Borda,  Bossut, 
Eytelwein,  Morosi,  &c.,  &c.,  among  the  former,  and  of 
Smeaton,  Robison,  Fenwick,  and  Banks  among  our  own 
countrymen ;  and  in  more  modem  times  by  Navier,  Ponce- 
let,  Morin,  Foumeyron,  &c.,  and  by  several  eminent  me- 
chanicians in  this  country. 

Of  the  several  classes  of  overshot,  breast,  and  under- 
shot wheels,  a  great  diversity  of  opinion  prevailed*     By 
Pitot  it  was  maintained  that  the  float  boards  of  undershot 
wheals  should  be  continued  in  the  line  of  the  radius.     By 
Deparcieux,  that  the  floats  should  be  inclined  to  an  angle 
of  15  or  more  degrees.     Bossut  was  of  a  contrary  opinion. 
Borda,  Bossut,  and  Robison  considered    that  the  maxi- 
mum velocity  of  the  wheel's  circumference  should  be  one 
third  of  the  velocity  of  the  current     Smeaton  made  the 
maximum  velocity  of  the  wheel  between  one  third  and  one 
half  of  the  current    Banks  difiers  firom  all  the  authorities. 
Navier,  Poncclet,  and  Morin  %  make  it  one  half,  whether 
the  floats  are  on  the  line  of  the  radius  of  the  wheel,  or 
curved.     Again :  as  regards  the  diameter  of  the  wheel,  it 
was  maintained  by  some  that  the  diameter  of  the  wheel 
should  never  exceed  the  height  of  the  fall,  and  by  others 
that  the  diameter  should  in  all  cases  exceed  the  height  of 
the  fall,  in  which  latter  opinion  Smeaton  coincides ;  for,  says 
he,  **  the  higher  the  wheel  is  in  proportion  to  the  whole 
<lescent,  the  greater  will  be  the  eflect.**     The  same  divers- 

*  Exporionccs  snr  los  Roues  Hydiuuliqnes  a  Aubes,  Planes,  et  svr  les 
Roues  Hydmuliques  a  Aujets^  1836.    Subsequent  experiments  haTe 
the  useful  effect  to  75  per  cent  of  the  abeolnte  expanditure. 


THIRD   EDITION.  *Xxiii 

ity  of  opinion  existed  relative  to  the  proper  number  of 
floats ;  Pitot  maintaining  that  the  number  of  floats  should 
be  equal  to  360**  divided  by  the  arc  of  the  circle  plunged 
in  the  chamiel,  and  Fabre  and  others,  that  the  number 
should  be  as  great  as  possible.  Bossut  found  the  best 
number  of  floats  to  be  forty-eight.  Smeaton  from  24  to 
40,  for  a  wheel  of  20  feet  diameter,  and  Poncelet  30  to  36 
floats  for  a  wheel  of  1 6  to  18  feet  diameter*. 

In  considering  the  action  of  the  water  on  the  vertical  or 
radial  floats  of  the  common  undershot  wheel,  M.  Poncelet 
was  of  opinion  that  one  of  the  causes  of  the  small  efiect 
produced  by  undershot  wheels,  arose  from  the  imperfect  ac- 
tion of  the  fluid  by  shocks  in  entering  the  wheel,  and  by 
gravity  in  quitting  it ;  whereas,  if  the  floats  were  so  con- 
structed as  to  admit  of  the  water  entering  and  quitting  the 
wheel  quietly,  the  effect  would  be  a  maximum. 

The  inclination  of  the  floats  has  long  been  a  favourite 
project  with  mechanicians,  but  in  so  far  as  our  own  expe- 
rience goes,  little  or  no  benefit  has  been  derived  from  that 
arrangement ;  and  in  this  opinion  we  are  confirmed  by  the 
experiments  of  Bossut,  who  found  the  effect,  at  different 
angles,  to  be  rather  disadvantageous  than  otherwise.  But 
M.  Poncelefs  curved  floats  produced  the  following  results. 

*  The  following  experiment  was  mode  in  the  year  1820,  by  the  editor  of 
this  work,  upon  two  water  wheels,  each  19  feet  4  inches  diameter,  and  6 
feet  in  width ;  one  wheel  had  40  floats,  and  the  other  48  floats,  and  tlie 
&D  12  feet  2  inches  in  height.  The  machinery  consisted  of  two  Avheels, 
Hid  two  pinions  of  cast  iron,  and  tAVo  pairs  of  4  feet  diameter  French  burr 
■tones. 

The  result  was,  that  the  wheel  with  40  floats,  ground,  in  31  hours,  359 
Imiheils  of  whent ;  and  the  wheel  with  48  floats  ground,  in  32  hours,  392 
hodielB  of  wheat.  Hence,  the  wheel  with  40  floats  ground  the  same  quantity 
of  wheat  (by  experiment)  with  1*43  per  cent,  less  water  than  the  wheel  with 
48  floftts.  It  was  also  proved  that  58*33  lbs.  of  water  ground  12*74  lbs.  of 
wheet  per  minute,  and  650*62  cubic  feet  of  water,  falling  one  foot,  ground 
one  Imhel  of  60  lbs.  weight  of  wheat,  for  the  wheel  with  40  floats. 


XXIV 


PREFACE   TO  THE 


Istly.  That  the  TnaximuTn  velocity  of  a  wheel  with  cuired 
floats,  was  0*55  of  the  velocity  of  the  stream. 

Qdlj.  That  the  dynamic  effect  for  small  falls  and  large 
openings,  and  0*65  for  large  falls  and  small  openings,  and, 
generally  speaking,  the  effect  of  the  wheel  with  the  curved 
float,  compared  with  the  effect  of  the  wheel  with  vertical 
floats,  was  as  0*60  to  0*50  of  the  power  expended. 

M.  Poncelet^s  researches  on  this  subject  have  placed  it 
upon  its  true  basis.  But  it  is  to  the  experiments  of  the 
committee  of  the  Franklin  Institution  *  that  we  are  indebted 
for  the  most  detailed  information  we  possess  on  the  subject 
of  water  wheels. 

The  great  defect  of  all  former  experiments,  is  the  small 
scale  upon  which  they  have  been  made. 

The  experiments  of  the  Franklin  Institution  were  made 
with  wheels  of  20,  16, 10,  and  6  feet  diameter,  respectively) 
and  all  the  conditions  of  friction  and  other  retarding  forces 
were  strictly  attended  to.  The  results  prove  that  the  maxi* 
mum  and  mean  effects  of  large  wheels  are  greater  with  mo^ 
derate  velocities  (double  what  Smcaton  assigned  as  the 
maximum  velocity  for  the  mean  circumference  of  a  water 
wheel).  The  maximum  and  mean  effects  are  diminished 
with  an  increased  velocity,  as  the  wheels  are  diminished 
in  diameter. 


*  20  foet  ^ameter  wheel  : 
128  Exi>crimeiit8 

Maximum 
effect. 

Correroond. 
veloaty. 

Mean  effect 

Coireapond. 
velodtj. 

•800 
•692 
•643 
•567 

5-48 

5-87 
5'8S 
7-59 

•784 
•609 
•562 
•484 

601 
5^73 
7^90 
8-18 

15  feet  diameter  wheel  : 
88  Experiments 

10  feet  diameter  wheel : 
180  Experiments 

6  foet  diameter  wheel : 
178  Experiments 

THIRD    EDITION.  •XXV 

In  the  case  of  vertical  water  wheels,  the  water  acts 
either  by  its  impulse  or  gravity.  But  with  horizontal 
wheels  with  inclined  or  curved  floats,  the  motion  is  pro- 
duced by  the  impulse  and  gravity  of  the  water  conjointly. 
The  experiments  of  Messrs.  Piobert  and  Tardy  •  on  several 
wheels  of  this  description,  in  the  south  of  France,  have 
given  very  feeble  results,  seldom  exceeding  one  fourth  of 
the  power  expended,  and  averaging  much  less.  The  reac- 
tion of  a  column  of  water  upon  the  curved  floats  of  a  hori- 
zontal wheel  has  been  found  to  bo  more  effective,  and  the 
recent  experiments  of  M.  Morin^  upon  the  Turbine  of 
Foumeyron  have  shewn  this  new  and  curious  machine, 
when  properly  constructed  and  moving  at  its  maximum  ve- 
locity, to  be  equally  effective  (if  not  more  so)  with  the  best 
vertical  wheels.  The  effect  of  the  reaction  of  a  column  of 
water  had  previously  attracted  the  attention  of  Euler  and 
Bernoulli  in  17^0%  and  a  machine  was  proposed  by  Euler 
in  1754,  upon  the  principle  of  the  steam  wheel  of  Hero 
of  Syracuse.  This  machine  was  further  improved  by  Man- 
noury  D'Hectot**,  who  constructed  several  in  the  neigh- 
bourhood of  Paris,  with  bent  tubes,  originally  suggested 
by  Euler.  The  theor\'  of  the  reaction  of  a  column  of 
water  against  the  sides  or  circumference  of  an  upright 
tube  when  allowed  to  flow  through  a  hole  or  pipe  fixed 
in  its  base,  has  often  been  investigated  by  philosophers. 
Daniel  Bernoulli,  in  his  Hydrodynamica  in  I788,  and 
John  Bernoulli,  in  his  Hydraulica,  and  in  the  St.  Peters- 
burg Transactions,  proposed  a  very  ingenious  and  elegant 
method  of  determining  the  impulse  of  a  column  of  fluid 
fidling  perpendicularly  upon  a  plane  surface  infinitely  cx- 

*  Ezp&iences  ewt  les  Roues  Hydrauliqucs  a  Axo  vertical,  Paris,  1840. 

b  Ibid.,  Turbines  Mctz,  1838. 

'  Memoires  de  rAcademie  de  Berlin. 

'  Journal  deB  Mines^  1813. 


*XZT1  PREFACE  TO   THE 

tended.  The  fonner  considered  the  curve  described  by 
every  filament  of  fluid  as  a  channel  in  which  a  body  moves, 
and  which  experiences  at  each  point  the  action  of  a  centri- 
fugal and  tangential  force,  which  varies  according  to  a 
given  law.  He  then  calculated  all  these  forces,  and  found 
that  the  impulsion  of  a  fluid  against  a  horizontal  plane  is 
equal  to  the  weight  of  a  column  of  fluid,  whose  base  is 
equal  to  the  section  of  the  fluid  vein,  and  whose  alti- 
tude is  equal  to  twice  the  height  of  the  fall  due  to  the  ve- 
locity of  the  fluid.  The  theory  was  afterwards  very  fully 
verified  by  a  series  of  experiments.  Tlie  question  of 
water  flowing  from  a  cylindrical  or  any  other  shaped 
vessel  was  also  treated  by  Madaurin  in  his  Fluxions,  pub- 
lished in  1742.  But  the  application  of  the  principle  of  re- 
action to  produce  motion  in  machines,  is  due  to  Segner  % 
professor  of  mathematic^^  at  Gottingen,  who  first  con- 
structed the  machme,  commonly  known  as  Barker's  miU\ 
The  celebrated  Euler  made  this  machine  of  Segner  the 
object  of  his  investigation,  in  a  paper  published  in  the 
Memoirs  of  the  Academy  of  Berlin,  in  the  years  17^0  and 
17^1f  and  shewed  that,  in  order  to  produce  the  greatest 
eflFect,  as  weU  from  the  pressure  as  from  the  centrifugal 
force  of  the  effluent  water,  it  was  necessary  to  curve  the 
horizontal  arms  or  tubes  of  the  machine,  so  that  the  aper- 
tures should  be  in  a  line  with  the  radius  of  the  wheeL  In 
1754,  he  again  turned  his  attention  to  the  subject,  and 
constructed  a  machine  with  two  systems  of  wheels,  the 
upper  wheel  or  cylinder  which  received  the  water  being 
fiixed,  and  the  lower  one  moveable  and  attached  to  a  ver- 
tical axis ;  the  water  then  flowed  from  the  upper  cylindrical 
to  the  lower  conical  wheel,  and  from  thence  through  twenty 
small  conical  pipes  fixed  into  its  circumference,  into  the  air, 

*  Exercitationes  hydraulicsB,  Qott  1  li^l. 

^  It  was  called  Segnersche  Wassenad,  in  Geamianj. 


THIRD   EDITION. 


^ 
¥ 


(d  the  machine  to  revolve  *.  Mather  de  la  Coiir, 
and  'W'arm^I^  proposed  to  introduce  the  column  of  water  from 
below  at  once  into  the  horizontal  arms ;  and  a  patent  for  a 
similar  application  of  this  principle  has  recently  been  taken 
out  in  Scotland.  As  regards  the  effect  of  these  machines, 
Dpiniona  are  various  ;  Banks  does  not  estimate  it  at  above 
one  third.  Waring  concludes  that  the  greatest  effect  will  be 
produced  when  the  reloeitv  of  the  orifice  is  half  that  of  the 
issuing  water,  and  that  this  effect  will  be  nearly  the  same 
88  that  of  a  well  constructed  undershot  water  wheel. 

Mr.  Ewart'  estimates  the  maximum  effect  to  he  consider. 
-ably  greater  than  the  same  quantity  of  water  applied  to  an 
undershot  wheel,  but  less  than  that  which  it  would  produce 
if  properly  applied  to  an  overshot  wheel.  In  i82i  M. 
Burdiu  invented  a  modification  of  Segner's  machine,  which 
he  termed  turbine  a  riacfion*.  It  received  the  water  in  the 
tipper  part  of  a  cylindrical  drum,  and  allowed  it  to  issue  at 
its  base  through  a  series  of  helical  channels  wound  round 
the  outer  surface  of  the  drum,  and  from  these  through 
three  pyramidal  pipes  issued  horizontally  into  the  atmo- 
(berc  This  machine  was  found  to  produce  an  effect  of 
65  to  75  per  cent,  of  the  power  expended. 

It  was  reserved,  however,  to  M.  Foumeyron  to  bring 
tlh«  turbine  to  its  present  perfection,  and  this  he  has  ac- 
iplished,   after  the  most  unremitting  perseverance  of 


*  Joamal  de  Bozier. 
>  TrsDnctioDa  of  the  American  Philosophical  Society  oF  Pbiladelphio. 

'  On  lie  roeasure  of  Moving  Force ;  VoL  H.  Memoirs  of  the  Literary 
and  FtiLlo«ophicBl  Society  of  Manchester,  1 808. 

*  Aniulcsdes  Mines,  Tom.  III.,  1828.  A  more  improved  tnaehino  of 
Una  doecnption  erected  by  M.  Burdiu  at  Pontgihuud  in  Fnuice,  called  a  Tur- 
Udo  1  £vacuittioD  alternative,  when  submitted  to  the  teat  of  llie  iiictioD 
lercr  of  IVony,  produced  an  useful  effect  equal  to  67  per  cent,  of  the  power 
rapcnded,  and  pcrfonaed  the  same  qusotity  of  work  with  ooe  third  of  the 
waMr  formorly  r«iuuod  by  n  horizoatftl  whe«l  worked  by  the  percusnoD 


•  •  • 


*XXY1U  PREFACE   TO   THE 

many  years  devoted  to  the  subject.  As  before  stated,  the 
turbine  consists  of  a  horizontal  wheel  with  curved  floats, 
which  are  set  in  motion  by  the  pressure  of  the  water 
issuing  from  the  centre  to  the  circumference,  or  vice  versd, 
and  which,  having  performed  its  office,  quits  the  floats  hori- 
zontally. But  as  the  problem  requires  that  the  water 
should  enter  the  wheel  without  shocks,  and  leave  it  with- 
out velocity,  a  peculiar  kind  of  construction  both  of  the 
wheel  and  floats  is  necessary,  and  it  is  the  practical  deter- 
mination of  the  curves,  derived  from  experience  alone, 
which  has  led  M.  Foumeyron  to  the  solution  of  the  ques- 
tion. Most  of  the  turbines  established  by  M.  Foumey- 
ron in  France  and  Germany  have  been  submitted  to  the 
investigations  of  M.  Morin,  and  the  results  have  so  far 
exceeded  the  expectations  of  men  of  science,  as  must 
eventually  lead  to  a  very  considerable  modification  in  hy- 
draulic engines  as  first  movers;  and  the  report  of  the 
Commissioners,  Messrs.  De  Prony,  Arago,  Gambey,  and 
Savary,  appointed  by  the  Royal  Academy  of  Sciences  at 
Paris,  on  the  experiments  of  M.  Morin,  entirely  adopts  his 
conclusions.  M.  Morin's  experiments  were  made  upon 
the  turbines  erected  at  Moussay,  Miilbach,  Lupine,  Inval, 
and  at  St  Blaise  \ 

^  The  first  scries  of  experiments  w»s  made  on  the  turbine  of  Moussay,  m 
1837.  Tbo  diameter  of  the  wheel  T«-as  *085  metres,  or  33^  inches;  the 
height  of  the  fall  was  7|  metres,  or  24  feet  8  inches;  and  the  number  of 
turns  made  bv  the  wheel  varied  from  76  to  240  per  minute,  according  to 
the  opening  of  the  sluice ;  the  relation  between  the  cffectiTe  and  theoretical 
expenditure  of  T«*ater  ii'as  0*910.  The  maximum  effect  corresponded  to  a 
Telocity  of  180  to  190  turns  per  minute^  and  the  useful  effect  mu  0*690, 
or  from  31  to  52}  honaes'  poTi-er;  but  at  velocities  of  140  and  230  turns 
per  minxite^  this  illation  varied  only  from  0*650  to  0*690  of  the  absolute 
power  expendeiU  or  a  variation  of  only  -jW^  thus  showing  that  the  effect 
of  the  wheel  was  not  altered  materially  by  variations  in  its  velocity. 
The  wheel  also  was  not  affected  when  working  submerged  in  tail  water. 

The  wheel  at  Miilbach  of  only  2  mecres^  or  6^  feet  diameter,  and  a  fidi 
of  3|  metit«k  or  about  ll^  fwc^  with  a  volume  expended  of  2|  cMc 


THlttD   EDITION.  'XlUt 

The  Sixth  Essay  relates  to  changing  the  velocity  in 
machinery  hj'  means  of  lathes,  alternating  pulleys,  alter- 

1,  yielded  a  useful  effect  of  91  ioreea'  power,  or  78  per  cent,  of  the 
BnditDTe.     In  this  cose  tho  number  of  revolutions  of  the  ivbcel  varied 
1 00  per  minute. 

I  Th*  turbine  of  Lepine,  with  a  fall  of  2  metrca,  or  fij  feel,  and  a  velocity 
Jfirami  60  to  100  revolutions  per  minute,  yielded  a  power  ofl2  horses, 
f  Filwily,  the  turbine  at  St.  Blaixe,  with  a  fall  of  108  metres,  or  354  feet, 
■od  n  wheel  nnder  22  inobes  diameter,  miulc  2300  turns  iu  a  minute,  and 
traoainitied  a  force  equal  to  40  horses. 
,  M.  Horin  concJudes  from  bis  experiments : — 
I  latly.  That  turbines  are  equally  adapted  to  great  as  to  small  falls  of 


Sdly.  That  they  are  capable  of  tronsmitlmg  an  useful  effect  equal  to 

O-70  to  0-78  of  the  absolute  power. 
3dly.  That  their  velocities  may  vary  very  considerably  from  tho  mftxi- 

cffpct,  without  differing  very  sensibly  from  it. 
-Hhly.  That  they  will  work  nearly  as  cffectuaUy  when  drowned  to  tho 
itb  of  one  or  two  metres,  as  when  free. 

SiUy,  Tliat  conBcquently,  they  will;  make  use  of  the  whole  of  the  fall 
leu  platod  below  the  level  of  estreme  low  water, 
ethly.  That  they  may  receive  variable  quantttiea  of  water  without  al- 

tbe  ratio  of  the  power  to  the  effect. 
And  if  to  thcee  properties  be  added  ^mplicity,  economy,  and  compnct- 
Ipgcther  with  the  facility  of  communicating  high  velocities  to  iuB' 
lery  without  the  intervention  of  wheels  or  pulleys,  the  turbine,  he 
^ra,  ought  to  rank  among  the  best  bydraiilic  machinery'  in  use. 
At  SL  Maur,  near  Paris,  four  turbines  have  been  erected  for  the  purpose 
gnodiiig  corn.  Each  turbine  is  3  feet  2  inches  in  diameter,  and  8  inches 
ibickueris  a""!  makes  50  revolutions  per  minute,  driving  10  pair  of 
ie«  3  feet  *  inches  in  diameter  at  the  rate  of  200  revolutions  per  minute, 
each  hirbliic  doing  tho  work  of  10  horses'  power. 

At  Corbeil,  about  16  miles  from  Paris,  M,  D'Arblay  has  recently  re- 
pkoi-d  twu  out  of  four  vertical  iron  wheels  upon  the  best  principles,  and  re- 
gihMWd  tliem  with  two  turbines  of  similar  diameters  as  those  at  St.  Maur, 
and  tluiT  ore  now  working  each  IU  pairs  of  stones  with  the  greatest  regu- 
.hrttj  and  satisfaction. 

For  farthw  information  on  this  snhject,  sec  Experiences  sur  !es  Roues  a 
rcrticn],   par  M.  Anliur  Morin,  Metz,  1838.      Also  Versuche  mit 
DemoDUKlen  Wosscrriiden  von    Herrn  Wedding   nnd  Herni   Carliczect, 
BetOn,  183*. 


*XXX  PREFACE   TO   THE 

nating  cones,  friction  wheels,  mules,  and  double  speeds,  as 
applied  to  cotton  spinning.  The  theory  of  mechanical 
motions  has  been  very  little  examined  until  recently.  Some 
of  the  early  writers,  such  as  Ramelli,  Bessoni,  Zonca,  &c, 
describe  the  various  continuous  or  alternate  motions  used 
in  machines;  but  these  motions  were  scarcely  classi- 
fied until  1794(,  when  Monge  produced  his  Elements  of 
Machines  for  the  use  of  the  Polytechnic  SchooL  It  was 
afterwards  treated  by  Hachette%  Lanz  and  Betancourt\ 
Ampere'  and  Borgnis*,  Whewell'  and  Willis'. 

In  the  Trait6  de  M6canique  of  Borgnis,  mechanical 
organs  are  divided  into  six  classes. — 1st  Receptors,  under 
which  are  classed  every  description  of  machine  moved  by 
the  power  of  animals. — ^Sdly.  Hydraulic  receptors,  such  as 
vertical  and  horizontal  wheels,  machinery  moved  by  the 
reaction  or  pressure  of  water,  or  by  heat,  vapour,  or  wind. 

Under  the  secondand  third  classes,  or  communicators  and 
modifiers,  are  machines  for  transmitting  and  modifying  mo- 
tion, such  as  toothed  wheels,  eccentrics,  indinedor  curvilinear 
surfaces,  chains,  levers,  pulleys,  wheel  eccentrics,  screws, 
cams,  &c«,  together  with  the  machines  for  producing  con- 
tinuous, or  variable,  or  alternate  motions.  The  fourth  daas 
comprehends  simple  supports  for  maintaining  vertical  or 
horizontal  axles,  and  rotative  supports  for  wiMTifaMTirng 
motions  of  translation  m  one  or  more  directions ;  and  under 
the  third  class  in  this  division  are  comprehended  toob. 
The  fifth  order  relates  to  r^rulalcNrs,  such  as  fly  wheels, 
governors  counter  weights,  horological  scapements,  ec- 
centric wheels,  curvilinear  motions,  friction  levers,  and 


^  Composauon  dc«  MMhines.  ISOS. 

^  Kmu  Air  k  Philowpbie  des  ScMnee^  ISas. 

*  Tndu^  dc  MMuq[Q«« 


THIRD    EDITION.  *XXX1 

nScal   pulleys   and  wheels,  (alluded  to  by  Buchanan.) 
The  sixth,  or  last  class,  termed  operators,  comprehends 
'ery  kind  of  machine    for    blowing  air,    for    agitating 
quids,  or  semifluids,  or  solids  j  for  compressing  substances 
r  means  of  rollers  and  presses,  or  for  stretching  or  ex- 
nding  metals :  again,  for  operating  by  friction,  such  as 
nding,    polishing    and    filing.        Fourth    sub-division, 
by    shocks,   such  as  hammers,  stampers,   pile  en- 
,  wedges,  &c.    And  under  the  last  or  fifth  sub-divi- 
tioa,  come  the  operators  by  separation,  such  as  rakes, 
scribbling  and  carding  machines,  knives,  chisels,  scrapers 
and  boring  tools. 
^H^  A  new  work,  however,  by  Professor  Willis,  has  just 
^Hppeared,   the  object   of  which,   to  use    his   own  words, 
^^phns  been  to  form  a  system  that  would  embrace  all  the 
^VuBmeDtary  combinations  of  mechanism,  and  at  the  same 
uptime  admit  of  a  mathematical  investigation  of  the  laws  by 
which  their  modifications  of  motion  are  governed.     1  have 
therefore,  says  he,  confined  myself  to  the  elements  of  pure  me- 
chanism, that  is,  to  those  contrivances  by  which  motion  is 
commonicated  purely  by  connexion  of  parts,  without  re- 
qniring  the  essential    intermixture  of  dynamical  effects. 
Instead  of  considering  a  machine  to  be  an  instrument  by 
means  of  which  we  may  change  the  direction  and  velocity 
of  a  git>en  moHon,  I  have  treated  it  as  an  instrument  by 
i  of  which  we  may  produce  any  relations  of  motion 
reen  two  jneces."' 
The  system  adopted  by  Professor  Willis  is  condensed, 
I  a  ^'nopticai  table  of  the  elementary  combinations  of  pure 
xhouism,  into  five  divisions  and  three  sub-divisions: 
The  first  class  comprehends  motion  by  rolling  contact, 
toothed  wheeb,   annular  wheels,  racks,  sectors, 
face  gearing,  hook  gearing,  and  wheels  in  general  for  pro> 
'  Willis's  Principlea  of  Meclianism,  1841. 


*XXXU  PB£FAC£   TO   THE 

ducing  constant  or  variable  velocities,  or  a  combination  of 
both. 

The  second  division  includes  motion  produced  by  sliding 
contact,  such  as  cones,  screws,  and  worms,  pin  and  slit  levers, 
spiral,  and  other  curved  surfaces,  under  the  different  cir- 
cumstances of  constant  or  variable  motion. 

The  third  division  shews  how  the  same  motion  can  be 
produced  by  wrapping  connectors,  such  as  guide  pulleys, 
gearing  chains,  curvilinear  pulleys,  and  fusees. 

The  fourth  division  includes  the  motion  produceable  by 
link  work,  such  as  cranks,  joints,  ratchet  wheels,  and  inter- 
mittent link  work. 

And  the  fifth  or  last  division  includes  reduplication  by 
means  of  tackle  of  ropes,  either  parallel  or  unparalleL 

The  aggregate  combinations  and  velocities,  and  adjust- 
ments of  machinery,  are  treated  with  that  ingenuity,  pre- 
cision, and  order,  which  might  be  expected  finom  the  au- 
thor. As  regards  the  practical  application  of  the  various 
motions  used  in  machinery,  we  need  only  adduce  the 
early  inventions  introduced  into  the  texdle  fabrics  by  Ark- 
wright  and  Cromptim,  Wyatt,  Hargreaves,  of  Watt,  of 
Boulton,  of  Huddart,  and  others  who  have  illustrated  the 
history  of  mechanical  inventions,  not  to  mention  invidi- 
ously inventors  and  men  of  science  who  in  modem  times 
have  carrioil  the  art  to  the  highest  perfi^^tion. 

The  Sewuth  Kssav  treats  of  the  framing  of  mill  work 
and  small  maohim^r}%  acconling  to  the  principles  of  Robi- 
sim  and  T>e<l^  Jd« 

The  obj^vt  \4r  framiugs  in  mill  work»  is  to  support  and 
uuuutain  th^^  diffV^n^nt  i>art$  i\f  machines  in  their  proper 
ainl  n^lative  vU^taiH^  $a^  that  all  the  wheels  shall  work  as 
$m\H>ihly  as  [¥^Wt\  and  witUvHit  shirks  or  vihraiicHis ;  for 
this  }HirtH^8?e  it  is  mn^Nssan  that  the  framing  be  made  in  con- 
f\>rmitY  to  the  strn^t^xi^t  ruW^j^  \rf  ;f^*ieiKv ;  that  is*  with  refer- 


THIRD    EDITION'.  'MXIIl 

QDce  to  the  composition  and  resolution  of  forces,  that  the 
resultants  of  these  forces  should  he  represented  by  ties  or 
struts ;  in  short,  that  all  pressures  should  be  so  distributed 
and  resisted  as  to  maintain  a  perfect  state  of  equilibrium 
throughout.  In  obtaining  a  knowledge  of  these  principles, 
it  is  necessary  that  we  understand  the  properties  of  the 
materials  with  which  we  have  to  deal  j  their  strength  and 
stress  in  all  positions,  their  durability,  and  their  powers  to 
resist  decay.  These  properties  will  be  found  in  our  table 
of  the  strength  of  materials,  and  it  is  on  the  judicious  distri- 
bation  of  these  materials  that  much,  if  not  the  whole  of  the 
art  of  the  mechanician  dejiends.  In  all  cases  of  tension,  to 
use  wrought  iron,  and  in  those  of  compression,  cast  iron  ; 
to  observe  the  proper  forms  best  suited  to  the  pressure  or 
tension  they  are  to  undcrfro,  and  to  avoid  as  much  as  pos- 
shle  the  use  of  framing  in  all  heavy  machinery,  availing 
lives  of  masses  of  materials,  such  as  stone,  brick, 
icrete,  or  sand,  in  all  cases  where  vibrations  or  shocks 
to  be  resisted.  For  although  cast  iron,  as  a  material, 
combines  the  advantages  of  stiffness,  strength,  and  dura- 
biH^>  and  the  facility  of  its  being  moulded  into  every  pos- 
sible form  suited  to  the  framing  of  small  raachinerj-,  yet  it 
is  occasionally  subject  to  break  by  unequal  contraction  in 
the  cooling,  and  by  blows  or  changes  of  temperature. 
Framing  of  wrought  iron  ia,  therefore,  much  used  in  marine 
steam  engines. 

The  Eighth  Essay,  although  not  in  the  original  edition 
of  Buchanan's  Essavs,  treats  of  the  geometrical  and  prac- 
tical methods  for  finding  the  centres  of  gravity  of  miU 
lis,  illustrated  by  examples  of  two,  three,  or  four  wheels 
red  upon  the  same  shaft.  This  subject  has  been  so 
iplv  illustrated  by  Dr.  .lamicson,  but  particularly  in  his 
Mechnnics  for  Practical  Men,   that  further  comment  is 

Irecssary. 
series  of  tables  of  square  and  cube  numbers  and  roots, 


peat   r 
^■leels 


•XXXIV  PREFACE   TO   THE 

taken  from  Hutton's  Course  of  Mathematics,  closes  the 
whole  of  Buchanan's  work. 

In  the  precedmg  ohservations  we  have  confined  our  at- 
tention to  a  hrief  outline  of  the  past  and  present  state  of 
our  knowledge  of  the  subjects  treated  by  the  Essays,  and 
an  imperfect  review  of  the  labours  of  those  to  whom  we 
are  so  deeply  indebted  for  the  knowledge  we  possess  of 
mechanical  science.  The  labours  of  Buchanan  are  con- 
fined to  the  development  of  a  few  elementary  principles 
connected  with  practical  mechanics,  excellent  in  them- 
selves, but  defective  both  in  the  extent  and  arrangement 
necessary  to  a  complete  system  of  mechanics. 

The  science  of  mechanics,  which  treats  of  the  equilibrium 
and  motion  of  solid  or  fluid  bodies,  and  which,  under  its 
various  divisions  of  statics,  hydrostatics,  dynamics,  and 
hydrodynamics,  comprehends  the  theory  of  action  and  re- 
action. Practical  or  technical  mechanics,  on  the  con- 
trary, treats  of  forces  as  realities,  and  machines  as  material 
objects,  capable  of  transmitting,  regulating,  or  modifying 
motion.  It  also  depends  on  a  multitude  of  facts  com- 
bined together,  and  establishes,  by  way  of  experiment, 
values  to  every  element  subservient  to  industry.  Further- 
more, it  determines  the  value  of  animate  and  inanimate  force, 
such  as  the  force  of  men  and  animals ;  the  force  of  gravity, 
such  as  weight,  water,  or  other  fluids ;  of  elastic  fluids,  such 
as  wind,  steam,  gas,  &c.,  all  of  which  forces  are  made  sens- 
ible through  the  agency  of  machinery.  By  machinery  we 
understand  an  assemblage  of  materials,  particles  or  parts 
susceptible  of  receiving,  communicating,  or  modifying 
motion.  A  machine  may  consist  of  a  simple  or  compound 
lever,  or  assemblage  of  levers,  revolving  on  a  centre,  such 
as  band  wheels  or  rollers,  or  any  of  the  mechanical  powers ; 
or  it  may  be  divided  into  three  parts, — the  parts  which  re- 
ceive, the  parts  which  transmit,  and  the  parts  which  com- 
municate or  perform   the  woA :   aU  these  motions  are 


THIRD    EDITION. 


•xxxv 


I 


I 


by  certain  reaistances  which  we  terra  passive,  such 
as  inertia  and  friction,  but  which  deduct  or  abstract 
from  the  absolute  force  in  proportion  to  the  perfection  of 
and  mode  of  applying  the  machine.  Machines  may  be 
employed  for  displacing  solid  or  fluid  masses,  for  changing 
the  forms  of  ductile  and  compressible  materials  by  pres- 
sure, for  separating  masses  of  solid  materials  by  friction, 
for  producing  changes  of  volume  in  solids  by  percussion, 
for  separating  solids  into  fragments  by  the  same  force,  for 
TniTcing  solids  together  by  penetration,  for  separating  fila- 
mentous substances  from  other  extraneous  substances  with 
which  they  are  interlaced,  and  for  rearranging  and  inter- 

I lacing  them. 
I  Whatever  be  the  nature  of  the  machine,  it  ought  to  be 
%o  combined  that  its  useful  effect  be  as  great  as  possible  ; 
that  it  should  be  as  simple  in  its  construction  as  possible ; 
that  its  parts  shoidd  combine  strength,  stifiness,  lightness, 
oniformity  of  action,  and  he  as  free  as  the  nature  of  the  re- 
Mfltance  will  permit  from  passive  resistance ;  that  it  should  act 
without  shocks  or  sudden  changes  of  motion  j  and  that  the 
comnmoication  between  the  power  and  resistance  should 
be  as  simultaneous  as  possible.  These  important  prin- 
ciples exact  an  intimate  knowledge  of  the  properties  of 
materials,  the  modes  of  transmitting  motion  in  all  its 
varieties,  of  contact  by  means  of  the  teeth,  cams,  and  other 
mured  surfaces,  by  bands  and  pulleys,  or  by  direct  or 
oblique  pressure.  Machines  are  the  implements  of  manu- 
fiujture,  a  word  which  applies  to  every  product  of  art 
which  is  made  by  machinery,  and  with  little  or  no  aid 
from  human  labour.  It  forms  a  separate  section,  or  rather 
a  science,  of  automatic  labour.  It  is  the  automatic  science 
which  bos  raised  our  country  to  its  present  elevated  posi- 
tion in  the  world,  as  displayed  in  its  cotton,  silk,  woollen, 
and  flax  manufactures ;  in  its  multitudinous  and  beautiful 

a  2 


•XXXVl  PREFACE    TO   THE 

machines  for  shortening,  multiplying,  and  even  dispensing 
with  the  labour  of  man,  evinced  in  the  construction  of  au- 
tomatic machines  for  creating  the  instruments  of  power 
whereby  the  elements  are  chained  to  perform  their  un- 
remitting toil, — whereby  the  powers  of  wind  and  water,  and 
steam  and  gas  are  rendered  subservient  to  our  uses,  and 
ere  long,  let  us  hope,  that  mysterious  power  of  electric 
magnetism,  which  seems  to  govern  alL  What  have  we 
not  witnessed  during  the  present  century  ?  If  we  turn  to 
the  triumphs  of  steam,  we  find  that,  whereas  the  duty  of 
the  pumping  engines  in  Cornwall  in  the  year  1808  was 
barely  equal  to  20  millions  of  pounds  of  water  raised  one 
foot  high  by  a  bushel  of  coals ;  in  1835,  the  duty  per- 
formed by  Mr.  Austen's  engines  at  the  Fowey  Consols  and 
Lanescot  mines,  with  an  80  inch  cylinder,  was  upwards  of 
125  millions  of  pounds  of  water  lifted  by  one  bushel  of 
coals  weighing  94  lbs.,  and  this  has  been  confirmed  more 
recently  by  the  valuable  experiments  of  Wicksteed  \  Thus 
carrying  out  the  ideas  and  principles  of  the  great  Watt,  so 
fully  detailed  in  his  patent  for  178^  and  in  the  works  of 
Robison\  Tredgold',  and  Farey*.  If  we  look  to  the 
marvels  which  have  been  effected  in  locomotion*,  both  on 
sea  and  land,  no  longer  subject  to  the  uncertainty  of  the 
elements,  the  untiring  machine  impels  the  mighty  fabric 
against  the  wind  and  waves,  annihilating  almost  time  and 
distance  between  the  New  and  Old  Worlds,  while  by  its 
stupendous  energies,  and  the  art  of  the  engineer,  dis- 
tances ae  no  longer  measured  by  space. 

*  Expenmental  Enquiry  concenimg  the  Rdatiye  Power  and  Useful  ESect 
produced  by  the  Cornish  and  Boulton  and  Watt  Pumping  Engine  and 
CyUudrical  Waggon-bead  Boi]«s.     1841. 

^  Robison^  Article  Steam  Engine. 

«  Treilgold  on  the  Steam  Engine,  2  toIs.     Weale,  1838-40. 

"^  Farey  H  Treatise  on  the  Steam  Engine* 

*  Comte  de  Pkanbour  s  Practical  Treatise  cm  Locomolhve  K^es,  1840. 


THIRD    EDITION.  'xXXvil 

Let  us  reckon  upon  the  future,  eays  M.  Arago,  iu  his 
istorical  eloge~of  James  Watt 
'  A  time  will  come  when  the  science  of  destruction 
shall  bend  before  the  arts  of  peace  ;  when  the  genius  which 
multiplies  our  powers,  which  creates  new  products,  which 
diffuses  comfort  and  happiness  among  the  great  mass  of 
people,  shall  occupy,  in  the  general  estimation  of  mankind, 
that  rank  which  reason  and  common  sense  now  assign  to  it. 
■  *'  Then  Watt  will  appem-  before  the  grand  jury  of  the  in- 
Btabitants  of  the  two  worlds.  Every  one  will  behold  him, 
with  the  help  of  his  steam  engine,  penetrating  in  a  few 
weeks  into  the  bowels  of  the  earth,  to  depths  which,  before 
his  time,  could  not  have  been  reached  without  an  age  of 
the  most  toilsome  labour,  excavating  vast  mines,  clearing 
them  in  a  few  minutes  of  the  immense  volume  of  water 
which  daily  inundates  them,  and  extracting  from  a  virgin 
Boil  the  inexhaustible  mineral  treasures  which  nature  has 
deposited  there.  Combining  delicacy  with  power.  Watt 
will  twist,  with  equal  success,  the  huge  ropes  of  the  gigantic 
cable  by  which  the  man-of-war  rides  at  anchor  in  the 
midst  of  the  raging  ocean,  and  the  microscopic  filaments 
of  the  aerial  gauze  and  lace.  A  few  strokes  of  the  same 
engine  wiU  bring  vast  swamps  into  cultivation,  and  fertile 
countries  will  also  thus  be  spared  the  periodical  returns  of 
deadly  pestilential  fevers,  caused  in  those  places  by  the 
beat  of  the  summer  sim. 

"  The  great  mechanical  powers  which  had  formerly  to  be 
songbt  for  in  mountainous  districts,  at  the  foot  of  rapid 
cascades,  will,  thanks  to  Watt's  invention,  readily  and 
easily  arise  in  the  midst  of  towns,  on  any  story  of  a  house. 
The  extent  of  these  powers  will  varj'  at  the  will  of  the  me- 
chanician ;  it  will  no  longer  deiMiud,  as  heretofore,  on  the 
moit  inconstant  of  natural  causes,  on  atmospherical  in- 
fluences  

"  Installed  in  ships,  the  steam  engine  will  exercise  a  power 

aS 


^XXXVm  PREFACE  TO   THE 

a  ImnilredBald  greater  than  the  triple  and  quadruple  ranks 

of  rovers and  bjr  the  hdp  of  a  few  bushels  of  coal, 

waa  win  Tanqnish  the  eLements ;  he  will  play  with  calms 
and  cootrarr  winds  and  storms. 

"  Lasdv :  The  steam  engine  drawing  in  its  train  thou- 
sands of  traTellerSy  will  ran  on  railroads  with  far  greater 
speed  than  the  smiflest  raoe-horse."* 

**  And,  in  condosion,  let  ns  quote  the  opinion  of  Sir  John 
HorscheL    On  the  importance  of  practical  mechanics  (he 
sap^  in  his  admirable  treatise  on  the  Study  of  Natural  Phi- 
losophy,) *'  Practical  mechanics  is  in  the  most  preeminent 
sense,  a  scientiJU:  art^  and  it  may  be  truly  asserted,  that 
ahnosi  all  the  great  combinations  of  modern  mechanism, 
and  many  of  its  refinements  and  nicer  improvements,  are 
creations  of  pore  intdlect,  grounding  its  exertion  upon  a 
moderate  number  of  elementary  propositions  in  theoretical 
mechanics  and  geometry.     On  this  head  we  might  dwell 
long,  and  find  ample  matter  both  for  reflection  and  wonder. 
But  it  would  require  not  volumes  merely,  but  libraries,  to 
cnumarale  and  describe  the  prodigies  of  ingenuity  which 
have  been  lavished  on  every  thing  connected  with  machinery 
and  engineering.     By  these  we  are  enabled  to  diffuse  over 
the  whole  earth  the  productions  of  any  part  of  it,  to  fill 
eveiy  comer  of  it  with  miracles  of  art  and  labour  in  ex- 
change for  its  peculiar  commodities ;  and  to  concentrate 
around  us,  in  our  dwellings,  apparel,  and  utensils,  the  skill 
and  workmanship  not  of  a  few  expert  individuals,  but  of  all 
who,  in  the  present  and  past  generations,  have  contributed 
their  improvements  to  the  processes  of  our  manufactures/'  ^ 

•  Tlje  annals  of  racing  record  sereiml  wonderful  feats  performed  by  race- 
bone^— Eclipse  once  ran  2  mOes  in  2  minntes,  and  on  another  occasion 
4  vBw  in  «  minntes  and  2  flecrads.  Fljing  Childera  ran  over  the  New- 
Miikec  covrae.  7420  yards,  in  7|  minutes.  Greyhounds  have  been  known 
ti»  m  aeariy  as  &8t  as  raoe-horses. 

*  IVe&MBarr  DisconrBe  on  the  Study  of  Natoial  Philosophy,  pages  63 
awlCf. 


THIRD    EDITION. 


ON    TOOLS. 


I 


The  subject  of  tools  has  been  so  amply  illustrated  bv 
Mr.  James  Nasmyth,  in  the  Appendix,  that  little  remains 
to  be  added.  By  tools,  we  understand  instruments  em- 
ployed in  the  manual  arts  for  facilitating  mechanical  ope- 
rations by  means  of  hammers,  pmicbes,  cbisels,  axes,  adzes, 
jilanes,  saws,  driUs,  files,  &c.,  by  means  of  percussion, 
penetration,  separation  and  abrasion  of  the  substances  ope- 
rated upon  ;  for  all  of  which  operations  various  motions 
are  required  to  be  given  cither  to  the  tool  or  to  the  work. 
In  handicraft  work  the  tool  receives  motion,  but  in  self- 
acting  or  automatic  tools,  motion  may  be  given  to  either. 
In  the  case  of  the  turning  lathe,  the  tool  remains  fixed, 
and  the  object  moves.  In  that  of  the  planing  machine, 
the  tool  may  remain  fixed,  or  be  made  to  move  accoi-ding 
to  the  duty  required  to  be  performed.  In  almost  all  the 
other  machines,  such  as  the  slotting,  the  key-grooving, 
the  punching,  the  drilling,  the  nut-cutting,  the  teeth  of 
wheels  cutting,  the  boring,  the  screw-cutting  machines,  the 
tools  receive  motion.  In  the  screw,  bolt,  and  nut  ma- 
chines the  tool  is  either  moveable  or  fixed.  The  use  of 
handicraft  tools  is  coeval  with  the  earliest  periods  of 
antiquity,  and  the  recent  researches  of  modem  travellers 
have  proved  the  ancients  to  have  beeu  acquainted  with 
almost  all  the  tools  now  in  use'.  The  potter's  wheel,  the 
axe,  the  chisel,  the  saw,  &c,  attest  the  perfection  to  which 
the  mechanical  arts  were  carried  by  the  Greeks  and  Ro- 
mans, and  subsequently  in  the  arts  of  turning  exhibited 
by  the  Dondi  family,  in  the  construction  of  their  clocks 
and   machines  for  spinning  silk*",  in  the    middle  of  the 

*  Muiners  and  Customs  of  the  Ancient  Egyptians,  by  Sir  Gardner  Wil- 
kmaon,  F.R.S..  1837. 
^  UiHoire  dee  Sciences  Matliomatiques,  par  Guillaumo  Libri,  Vol.  I.,  1 838. 

a  4. 


•"xl  PREFACE   TO   THE 

ISth  century  in  Italy,  and  afterwards  by  Bessoni%  De  la 
Hire^  De  la  Condamine%  Grand  Jean^  Plumier,  and 
Morin*.  The  three  plates  of  Bessoni  shew  the  different 
modes  of  turning  and  cutting  screws  of  all  sorts  of  fancy 
work.  De  la  Hire  shews  how  all  sorts  of  polygons  may 
be  made  by  the  lathe,  and  Condamine  shews  how  a  lathe 
may  be  made  to  turn  all  sorts  of  irregular  figures  by  means 
of  tracers  moved  over  the  surface  of  models  and  sculptures, 
medals,  &C,  and  this  is  perhaps  the  first  idea  of  the 
machine  called  the  Tour  a  Portrait 

The  work  of  Plumier  enters  most  extensively  into  the  art 
of  turning,  for  he  shews  the  construction  of  the  lathe  and 
its  difierent  parts,  the  art  of  making,  hardening,  tempering, 
and  sharpening  tools,  the  different  kinds  of  motions  which 
may  be  given  to  the  lathe  by  means  of  wheels,  excentrics, 
and  models,  and  the  difierent  inventions  relative  to  works  of 
art  which  have  been  performed  by  the  lathe,  among  which 
mav  be  mentioned  the  moveable  or  slide  rest  In  the  com- 
mon  rest  which  supports  the  tool,  the  idea  of  fixing  the 
tool  and  pushing  it  in  the  direction  of  the  parallel  bed  of 
the  lathe,  so  as  to  cause  the  tool  to  traverse  the  work  pa- 
rallel to  it,  must  have  been  obvious,  and  as  this  could  have 
been  easily  effected  by  means  of  the  screw  and  handle,  it 
required  little  ingenuity  to  carry  out  the  idea  to  its  fullest 
extent,  by  constructing  a  rest  to  allow  of  the  slide  traversing 
the  horizontal  or  vertical  plane  in  any  direction.  The 
machine  described  by  Plumier  is  neither  more  nor  less 
than  the  slide-rest  and  planing  machine  combined:  it 
consists  of  two  parallel  bars  of  wood  or  iron  connected  to- 
gether at  both  extrt^mities  by  bolts  or  keys  of  sufficient 

•  TKecUnim  Mftchinarask  15S!2« 

^  MnehiiM^  AppnMiTf«$  p«r  rAcmdemie*  1719, 

*  IhKk  1733. 

^  MiichiiM>«  Ap|iiottTf««  )i«r  rAc«de«ue«  1733. 


C«] 

Lint 

or 
for 
by 
bai 


THIRD   EurrioM.  •ili 

vridth  to  admit  of  the  article  required  to  be  plaiied ;  a 
moveable  frame  being  placed  between  the  two  bars,  and 
motion  being  given  to  it  by  a  long  cylindrical  thread,  is 
capable  of  giving  motion  to  any  tool  which  may  be  put 
into  the  sliding  frame,  and  consequently  either  causing  the 
rew,  by  means  of  a  handle  at  each  end  of  it,  to  push  or 
aw  the  point  or  cutting  edge  of  the  tool  either  way.  If 
also  motion  be  given  to  the  tool  by  means  of  guides  upwards 
or  downwards,  it  is  evident  that  any  kind  of  reticulated 
form  can  be  given  to  the  work,  as  in  the  machine  described 
by  Plumier,  which  was  intended  for  ornamenting  the 
handles  of  knives,  and  which  is  called  by  Plumier,  Machine 
,d  mnnvfie  de  Coutemi  d" Angleterre',  from  its  ha^ong  been 
English  invention.  The  Machine  d  Conneler  de- 
scribed by  Bergeron '',  a  mode  of  grooving  columns,  is  ])ro- 
bably  derived  from  the  same  sowce,  from  its  resemblance 
to  the  English  machine.  We  have  given  a  plate  and  de- 
acription  of  Nicolas  Eorq's  machine  in  Plate  45  of  the 
present  work,  and  we  have  a  drawing  of  a  similar  machine 
which  was  used  in  (Jermany  many  years  back.  The  origin 
of  the  planing  mac-hine,  in  more  recent  times,  is  said  to 
hare  arisen  from  the  grooving  or  fluting  of  the  drawing  roU- 
ere  used  in  cotton  machines  shortly  after  the  introduction 
of  Arkwright's  inventions.  The  patent  of  Sir  Samuel 
Bentham"  in  1793,  for  various  new  methods  for  working 
WfKxl,  metal,  and  other  materials,  certainly  contemplates 
the  working  of  tools  similarly  to  the  tools  employed  in  the 
planing  macliine.  The  patent  comprehends  giving  all 
sorts  of  motion  to  tools,  and  the  patent  of  Joseph  Bramah'*, 
taken  out    in    1 802,    was  for  machinery   for  producing 


■  See  pngce  1.55,  15B,  anil  Pktes  5+,  5.5,  5fl,  Plumier  I'Art  de  Toumer. 
iFuia,  1754. 

>  Mnroel  du  Toumeur,  Paris,  1816. 
'  Repcrtorj'  of  Arte,  1793.  Vol.  X. 
•  Bq>«irMirf  of  Aftt,  1802. 

a  5 


♦xlii  PREFACE   TO   THE 

straight,  parallel,  and  smooth  surfaces  and  other  materials 
requiring  truth,  in  a  manner  more  expeditious  and  perfect 
than  can  he  performed  hy  the  use  of  axes,  screws,  planes, 
and  other  cutting  instruments  used  hy  hand  in  the  usual  way. 

Billingshy%  of  Birkenshaw,  took  out  a  patent  in  1802, 
for  horing  cylinders  in  a  vertical  position,  although  hori- 
zontal machines  had  their  advantages.  The  horing  of 
large  cylinders  hy  horizontal  machines  had  long  heen 
practised  hy  Smeaton,  Wilkinson,  Walker,  Darhy,  and 
Boulton  and  Watt,  and  at  Butterley  and  other  great  iron 
works ;  hut  it  was  only  within  the  last  few  years  that  the 
vertical  horing  machines  came  into  use. 

As  respects  the  introduction  of  the  first  planing  ma- 
chines which  have  heen  used  during  the  present  century, 
opinions  are  at  variance.  Messrs.  Fox,  of  Derhy,  the 
eminent  tool  makers,  state  that  the  first  machine  em- 
]doyed  for  this  purpose  was  constructed  hy  Mr.  Fox, 
senior,  in  the  year  1891,  for  the  purpose  of  planing  the 
wrought  and  cast  iron  hars  used  in  the  lace  machines. 
The  machine  was  capaUe  of  planing  an  article  10  feet  6 
inches  in  length,  22  inches  in  width,  and  12  inches  in 
depth ;  others  give  the  credit  of  the  invention  to  Man- 
chester, and  we  ourselves  put  in  our  claim  for  constructing 
a  planing  machine  with  a  moveable  bed,  urged  by  an  end- 
less screw  and  rack,  and  furnished  with  a  revolving  tool, 
so  early  as  1820,  having  several  years  previously  employed 
the  principle  for  grooving  and  planing  parallel  bars. 

Mr.  Bramah,  in  1811,  employed  the  revolving  cutter  to 
[dano  iron.  Mr.  Clement  ^  states  that  he  made  a  planing 
machine,  for  planing  the  sides  of  weaving  looms  and  the 
triangular  bars  of  lathes,  previously  to  1 820.  He  afler- 
wants  ixmstruotoit  a  beautiful  machine  for  planing  large 
and  small  wv>rk  with  the  sn^^eatest   accuracv.      The  bed 

«  RepertiMnr  of  Aft^  Vol.  lU  1^0^ 

^  ieih  tad  i$di  Y«4wuM«  ciT  a»  Ti«MactiQ«s  ^ 


^^  ant 
Btnl 


THIRD    EDITION.  •xliu 

■moTed  on  rollers,  and  the  tools  cut  both  ways.  The  beau- 
tiful work  executed  by  this  tool,  for  Mr.  Babbage's'calcu- 
Uting  machine,  evinces  the  perfection  of  its  performance. 
It  is  thus  by  the  aid  of  automatic  tools  that  we  are  enabled 
to  produce  the  greatest  precision  and  identity  of  parts  in 
machinery,  which  could  never  before  be  attained  by  ma- 
chines made  by  hand  labour ;  and  it  is  hoped  that,  ere 
long,  the  cbisel,  the  file,  and  the  grindstone  will  be 
banished  from  the  factorj-,  and  that  nicety  of  parts  and 
uniformity  and  silence  of  action,  blended  with  the  science 
of  construction,  will  eventually  supersede  the  expensive 
and  imperfect  construction  of  the  handicraft  system. 
We  might  enlarge  upon  this  subject,  by  detailing  the 
iture  and  properties  of  the  materials  required  for  tools  ; 
the  forging,  hardening,  and  tempering  of  them  ;  the  velo- 
cities at  which  they  should  be  made  to  move  through  the 
difierent  materials,  such  as  woimI,  iron,  brass,  copper,  and 
tin.  We  might  ^ve  the  principles  of  the  action  of  the 
different  machines  employed  to  produce  different  effects  ' ; 
but  we  have  exceeded  our  limits,  and  it  only  remains  to 
express  our  great  obligations  to  the  several  gentlemen  who 
have  so  liberally  assisted  us  on  the  present  occasion.  To 
Professor  Willis,  for  bis  article  on  the  Teeth  of  Wheels. 
To  Mr.  James  Nasmyth,  for  his  Paper  on  Tools,  and  his 
numerous  and  beautiful  drawings  of  the  tools  which  bear 
the  name  of  Nasmyth  and  Gaskell.  To  the  late  lamented 
Mr.  I-'rancis  Bramah,  we  owe  the  original  drawing  of 
the  first  slide  rest  of  his  father,  in  1794,  the  work  of 
the  late  Mr.  Maudslay,  and  it  is  yet  in  use ;  and  also 
for  the  drawing  of  the  lathe  for  turning  spheres.  To 
Mr.  FairbaJm,  for  his  plate  riveting  and  punching  machine, 
and  for  his  advice  and  assistance  on  several  occasions.  To 
Mr.  Wliiiworlh,  for  the  information  we  have  derived  fr«m 

'  EipcriineiiM  of  M.  Morin,  on  tie  Measures  of  the  Dynamic  EtTeeta  of 
MinniU)  ttnd  Animal  Power,  and  on  Machines  in  general. 


♦xliv  PREFACE   TO   THE   THIRD   EDITION. 

his  various  pamphlets  on  plane  metallic  surfaces,  and  on  the 
proper  mode  of  preparing  them ;.  likewise,  on  an  miiform  sys- 
tem of  screw  threads '.  To  Messrs.  Fox,  for  their  screw- 
cutting  machine,  and  other  information.  To  Messrs. 
Benjamin  Hick  and  Son,  of  Bolton,  for  the  liberal  assist- 
ance of  the  drawings  for  the  plates  which  bear  their  name ; 
and  we  take  this  opportunity  of  noticing  their  ingenious 
machine  for  cutting  the  teeth  of  the  largest  sized  wheel 
used  in  mill-work,  and  their  mandril  for  holding  rings ; 
their  steel  belts,  as  a  substitute  for  leather  bands,  are  used 
very  successfully.  To  Mr.  Francis  Lewis,  of  Manchester, 
we  are  equally  indebted  for  the  drawings  of  the  different 
machines,  placed  by  that  gentleman  at  our  disposal.  To 
Messrs.  Maudslay  and  Field,  for  the  liberal  present  of  the 
drawing  of  their  self-acting  punching  machine,  by  which 
accuracy  is  insured  in  the  heretofore  neglected  art  of  boiler- 
making  i  it  is  one  among  the  proofe  of  the  high  state  of 
excellence  to  which  those  gentlemen  have  brought  the  me- 
chanical arts  in  this  country.  A  table  of  references,  and  an 
ample  description  of  the  different  tools,  by  our  assistant, 
Mr.  George  Pinchbeck,  will,  we  trust,  explain  the  different 
details.  With  respect  to  the  plates,  it  is  suflRcient  to  state, 
that  they  are  engraved  by  Lowry,  a  name  too  well  known 
to  need  further  comment.  The  liberality  with  which  the 
whole  has  been  got  up  by  its  spirited  publisher,  will,  it  is 
trusted,  be  acceptable  to  the  public. 

■  On  an  Uniform  System  of  Screw  Threads.     8vo,  1841. 


We  are  indebted  to  the  late  Thomas  Tredgold  for  whatever 
is  known  of  the  life  of  Buchanan.  It  was  furnished  by  his 
friends,  and  though  brief,  the  life  of  a  man  of  genius  is 
always  interesting. 

Robertson  Buchanan  was  bom  on  the  I4th  of  July^ 
1769,  at  Glasgow,  and  was  connected  by  birth  with  some 
of  its  principal  citizens.  His  father  was  nephew  to  Neil 
Buchanan,  who,  in  the  year  I7G8,  represented  Glasgow 
m  Parliament  j  and  his  mother  was  daughter  of  Arthur 
Robertson,  who  for  many  years  was  chamberlain  of  that 
city.  Buchanan  lost  his  mother  at  his  birth,  and  his 
Enther  when  he  was  only  fifteen.  His  father  had  not  been 
fortunate  in  business,  and  the  son  was  left  unprovided  for, 
but  lie  had  already  shewn  some  talent  for  drawing  and  me- 
chanics, whifh  induced  his  maternal  uncle  to  place  him 
with  a  house-carpenter  at  Glasgow.  The  genius  of  Bu- 
chanan sought  its  native  field  in  a  short  time,  for  we  after- 
wards find  him  working  with  a  millwright,  and  subsequently 
crossing  the  border  for  London.  After  a  short  time  he 
quitted  the  metropolis,  returned  to  Glasgow,  and  com- 
menced business  there  as  a  millwright;  in  the  year  1791 
he  gave  it  up  to  take  the  management  of  the  new  cotton-mill 
then  building  at  Rothesay  in  the  Isle  of  Bute.  There 
he  invented  his  pump  which  is  not  liable  to  choke,  and 
for  which  he  obtained  a  patent  in  the  year  1796.  In  the 
same  year  he  wTOte  some  papers,  which  were  published 
Repertory  of  Arts  and   Manufactures  j — one  on 


•xlvi  LIPE   OF   BUCHANAN. 

the  improvement  of  cattle  mills,  another  on  preventing 
carding  machines  from  injuring  the  health  of  those  em- 
ployed to  attend  on  them. 

He  left  Bute  in  the  year  1801,  much  impaired  in  health 
by  the  anxiety  of  a  responsible  situation  in  a  losing  busi- 
ness, and  returned  to  London,  with  a  view  of  deriving 
some  benefit  from  the  pump  he  had  invented ;  but  in  this 
he  never  succeeded.  He  was  introduced,  however,  to 
Count  Rumford  and  Professor  Pictet,  by  whom  his  atten- 
tion was  directed  to  the  heating  of  rooms,  and  in  the 
year  I8O7,  he  published  an  "Essay  on  Warming  Build- 
ings by  Steam.**  He  had  previously  been  engaged  in 
preparing  the  "  Essay  on  the  Teeth  of  Wheels,**  but  when 
a  considerable  part  of  it  was  printed  ofi^,  an  unfortunate  oc- 
currence to  the  printer  and  publisher  delayed  the  publica- 
tion until  the  year  1808.  In  the  year  1810  he  published 
his  work  on  heating  buildings  in  an  improved  form,  with 
the  title  of  "Practical  and  Descriptive  Essays  on  the 
Economy  of  Fuel  and  Management  of  Heat.**  In  the 
year  1814  appeared  the  Six  Essays  on  Mill  Work,  which, 
with  that  on  the  Teeth  of  Wheels,  constitute  the  present 
work. 

In  the  year  181 6  he  published  a  practical  essay  on  pro- 
pelling vessels  by  steam,  a  work  fiill  of  new  views  and 
principles  in  that  most  important  art,  and  the  commence- 
ment of  a  new  era  of  civilization  in  the  annals  of  society  at 
large.  He  also  contributed  the  articles  "  Cotton-spinning  ** 
and  "  Arkwright  **  to  the  Edinburgh  Encyclopsedia,  besides 
several  papers  on  less  important  subjects. 

He  died  in  the  47th  year  of  his  age,  at  Creech  St.  Mi- 
chael,  in  Somersetshire,  on  the  22d  of  July,  I8I6. 

He  was  a  man  of  amiable  character,  with  a  strong 
sense  of  religious  and  moral  duty,  and  was  greatly  re- 
spected by  all  that  knew  him.  His  knowledge  in  mecha- 
nics was  very  extensive.      He  was  a  close  and  accu- 


LIFE   OF   BUCHANAN.  •xlvU 

rate  observer,  extremely  assiduous  in  collecting  every  fact 
or  experiment  which  came  under  his  notice,  and  he  was 
unquestionably  one  of  the  few  practical  men  who  have 
shewn  inclination,  or  sought  leisure,  to  reason  on  &cta  in 
general  with  accuracy  and  judgment,  aad  always  with  a 
view  of  rendering  the  information  thus  acquired,  an  avail- 
able source  for  unforeseen  emergencies.  Buchaaan  was 
happy  in  the  choice  of  popular  subjects,  and  he  fully  com- 
pensates  for  want  of  system  in  haadling  them  by  the  va- 
riety and  utility  of  particulars  no  less  interesting  than 
abundant^  whether  learned  from  his  predecessors,  or  de- 
rived from  contemporaries. 


ANALYTICAL  TABLE 


or  THS 


CONTENTS  OF  THE  WHOLE  WORK. 


Art.  Pig» 

OSNSRAL  PbBFACB y 

Preface  to  third  Edition vii 

Life  of  Buchanan *xly 

ESSAY  I. 

On  the  teeth  of  wheels,  comprehending  pnnciples  of  their 
i^plication  in  practice  to  mill-work  and  other  machinery. 

General  definitions  of  wheels  and  pinions,  trundles  and  teeth, 
cogs,  leaves  and  staves 1-4  1 

Of  the  line  of  centres,  and  the  proportionate  radii        .         .     6,  7  2 

Of  proportional  drdes  or  pitch  lines,  and  real  radii      •         .     8, 9  B 

CHAPTER  I. 

Of  the  principles  of  the  confignration  of  the  teeth  of  wheels      —  4 

Of  the  proper  formation  of  the  teeth  of  wheels   .         .         .10  4 

Of  one  wheel  conducting  another  as  if  they  simply  touched, 

or  their  pitch  lines  have  in  every  part  of  their  revolution 

equal  velocities II  5 

Notes  illustrative  of  peculiar  cases,  and  of  the  fundamental 

proposition —  6 

Demonstration  that  the  pitch  lines  have  in  corresponding 

places,  equal  velocities 12  6 

Definitions  that  the  epicycloid  ^ves  the  property  to  wheels 

whose  pitch  lines  shall  have  in  corresponding  places  equal 

velocities 13  7 

The  generating  circle  of  the  epicycloid       .         .         .         .14  7 

Of  the  exterior  epicycloid 15  8 

Of  the  interior  epicycloid 15  8 

Girollaries. — 1.  Determining  the  points  through  which  to 

trace  tbe  ^ncycloidal  curve 10  8 


XIV  CONTENTS. 

Alt. 

2.  The  generatbg  circle  revolTiog  within  the  dicle  of  its 
base 17  9 

d.  The  generating  circle  touching  the  drcomference  of  its 

base 18  10 

4.  Of  the  generation  of  the  figure  by  means  of  three  circles 

moTeable  round  their  centres  only 19  11 

5.  Mechanical  methods  of  efiecting  this  by  three  circles       .20  12 

6.  Mechanical  methods  by  two  circles  for  determining  the 
best  figure  which  can  be  given  to  the  teeth  of  wheels, 

when  the  pinion  shall  be  a  trundle  composed  of  staves     .       21  14 

Properties  of  the  epicycloid  both  curious  and  scientific  .22  16 

Rules  for  finding  the  lengths  of  epicycloidal  curves,  and  the 

areas  they  inclose 28  16 

Halley's  rule  for  the  area  of  all  cycloids  and  epicycloids       .23  17 

CHAPTER  II. 

Of  the  application  of  the  principles  of  the  configuration  of 

the  teeth  of  wheels 24  18 

Practical  explanation  of  the  epicycloid  curve       .         •         .24  18 

Section  I. — Of  spur  gear 25  18 

Of  the  wheel  and  trundle 26  18 

To  find  the  figure  of  the  teeth  when  the  staves  are  indefi- 
nitely small  27  19 

To  find  the  figures  of  the  teeth  of  the  wheel,  when  the  staves 

of  the  trundle  are  cylinders  of  a  finite  diameter        .         .       28  21 

To  describe  the  teeth  ^f  a  wheel  for  a  trundle  by  means  of 

circular  arcs 29  23 

Of  the  wheel  and  pinion. — To  find  thefi^re  of  the  teeth  and 
leaves  of  a  wheel  and  pinion^  when  that  part  of  the  teeth 
and  leaveSy  which  lies  within  their  respective  proportional 
circles  are  straight  lines  directed  to  the  centres  of  these 

circles 80  24 

Remarks. — On  friction 82  27 

On  what  it  depends 83  28 

Friction  of  metal  teeth ZZ  29 

Friction  at  the  line  of  centres,  and  receding  from  it     .         .83  30 

Rule  more  general   of  easier  application  for  describing  teeth 

than  that  of  Camus 34  30 

Demonstration  that  a  pinion  of  10  leaves  may  be  moved  uni- 
formly by  a  wheel  of  209  teeth 36  32 


CONTENTS. 


A  tnindle  with  less  than  8  staves  cannot  he  moTed  uniformly 

hy  a  wheel  with  any  number  of  t«etb  whatever 
The  qticvcloid  necessary  on  conductors  only,  whether  wboel 
or  pinion      ......... 

Stares  prefcr«blo  to  teeth,  if  but  few  in  the  pinion 

Advantage  of  a  small  trundle  over  a  pinion 

Wheels  of  cast  iron  ....... 

In  sniail  trundles  of  cast  iron,  toeth  are  preferable,  lim-iiig 

their  a4;ling  porta  of  the  figure  of  a.  stave 
Smple  method  of  describing  teeth  to  resemhle  staves  for  the 
conducted  wheel  or  pinion     ...... 

Method  of  calculating  the  real  radius  when  the  wheel  is  the 

conductor 

^^  Hetbod  of  calculating  the  smallest  real  radius  which  a  wheel 

^^L    adapted  to  a  tnmdie  can  have 

^^Blethod  of  calculating  the  smallest  real  radius  which  can  be 
^H^    given  to  a  wheel  adapted  to  the  leaf  of  a  pinion 
■        Advantages  of  long  teeth  over  short  ones  shown  in  case  of 
Asctnre  and  also  in  friction    ...... 

BnJe  to  determine  [he  length  of  the  teeth  of  wheels     . 
Obaemitions  on  the  preceding  rule    ..... 

Of  the  internal  pinion,  and  the  cases  in  which  it  may  be 
adopted  with  advantage  ...... 

IQuatmtian  tliat  it  has  less  friction  than  the  external  one 
Showing  also  that  upon  this  principle  bevelled  wheels  have 
less  friction  than  external  spur  wheels    .... 

Of  the  rack  and  pinion     ....... 

Should  be  made  upon  the  principles  of  spur  gear 
^^■jKgDfe  representing  the  teeth  of  a  rack  and  pinion  fonned 
^^B   for  cases  in  which  great  weight  is  attached  to  the  rock 
^^Hcorrect  eonatructiou  for  the  rack  and  pinion 
^^^jf  cthod  of  giving  durability  to  the  teeth  of  the  rack,  and  lind- 
iug  the  real  nidiiis  of  the  pinion      ..... 

Of  the  farm  of  the  face  of  tlie  teeth  of  the  rack  when  it  im- 
pels the  pinion 

Sbctioh  n. — Of  hovel  gear 

]t>  action  represented  by  cones  rolling  on  the  surfuce  of  each 

other 

K^rmnu  illustrating  their  motions 

Kzpbnstions  of  these  revolutious  of  cones  .... 


XVI  CONTENTS. 

Alt. 

How  the  epicycloid,  which  giTes  the  teeth  of  bevel  gear,  is 

generated 56  5B 

Illustration  of  the  spherical  epicycloid        .        ...      56  54 

Practical  method  of  laying  down  the  lines  necessary  to  the 

right  construction  of  boTol  gear 57  55 

To  determine,  firstly,  the  diameter  at  the  pitch  line  of  the 
wheel;  and,  secondly,  the  length  and  breadth    of  the 

teeth 57  56 

Mode  of  drawing  the  section  of  the  pinion  ...       57  57 

Specific  definitions  and  illustrations  for  moving  one  another 

uniformly 58  57 

1st,  When  the  wheel  drives  the  pinion,  having  the  acting 
faces  of  the  teeth  spherical  epicycloids ;  2dly,  when  the 
teeth  of  the  pinion  are  staves,  or  are  formed  to  act  as  staves ; 
ddly,  when  the  face  of  the  tooth  is  a  spherical  involute  of 

a  circle 58  58 

A  new  and  general  method  of  describing  these  curves  for 
wheels ;   spur  wheels,  racks,  &c.,  being  only  particular 

cases  of  its  application 59  58 

Developement  of  the  cone  teeth  upon  a  plane     ...       60  58 

Patterns  for  drawing  these  teeth 61  59 

Rules  for  regulating  the  limit  of  the  pitch,  and  breadth  of 

the  teeth 61  60 

CHAPTER  III. 

Carnprisififf  a  series  of  articks  having  direct  application  to 
mill'tDork^  and  the  teeth  of  wheels, — Professor  Robison's 
mode  of  forming  the  teeth  of  spur  wheels        ...       62  62 

Properties  of  involute  teeth  in  wheel-work  ...      63  63 

Supplementary  observations  on  the  alignments  of  Professor 

Robison,  Camus,  and  Dr.  Young 64  64 

Dr.  Young's  remarks  on  the  friction  of  bevel  gear  wheels, 

extracted  from  the  1st  vol.  of  his  Natural  Philosophy       .       65  65 

Dr.  Young's  opinion,  by  letter  to  Buchanan,  on  friction,  and 
the  forms  best  suited  for  teeth 

Supplementary  definitions  of  geometrical  figures 

Geometrical  constructions  of  plane  figures  . 

Definitions  of  geometrical  figures  resumed 

Definitions  of  some  terms  in  science  and  mechanics 


66-71 

66 

72-82 

71 

82-85 

73 

85, 86 

75 

87-91 

75 

mustmtions  of  the  term  power  as  used  in  mathematics  ;  of 
force — momentum         ,..,.,  02-97 

^^H  Of  moclianic&l  power  ...  .93 

^^■IhstiDctioa  between  the  nicsaure  of  power  and  the  meHsoro 

^^P    of  effect OS 

^^GnnmeroUon  of  works  on  mi!!- work  and  mecliaiiics,  useful  in 

fitud\iag  Buchanan's  kboufs 99 


CHAPTER    IV. 

i  p»ctic&l  inquir)'  respecting  the  Btrength  and  durability  of 
e  teeth  of  wheels  used  in  mill-work     .... 

k  On  the  Htrenglh  of  the  teeth  of  wheels  in  relation  to  the  re- 
islnnce  they  have  to  overcome       ..... 

Peneral  observobons  on  the  wbecl-work  of  mills 

e  mcaanrc  below  which  the  diameter  of  wheels  ought  not 
f  to  be  reduced        ........ 

piEthods  hy  wliich  a  saving  of  power  has  been  obtained 
e  praedcot  limits  of  a  fine  pitch,  ami  also  of  breadth 
indpleA  of  proportioning  the  teeth  of  wheels. — PBoroai- 
Tio»  I.  Tie  ttrenytk  of  any  piece  of  tivJ>ery  or  metal,  wAose 
tertian  it  a  reeiangle,  i*  in  direet  proportion  to  the  hreadth, 
omd  at  the  tquare  of  the  depth         ..... 

The  strength  of  tho  teeth  of  wheels  moving  at  the  some  velo- 
city and  under  the  same  circumstances,  is  directly  in  pro- 
portion to  their  breadth,  and  as  the  square  of  the  thick- 

ne« 

PaoPOSiTioir  n.—If  any  force  be  applied  latercdly  to  a  lever 
or  beam,  the  tirett  upon  any  place,  is  directly  as  the  force 

and  ittditlancc  from  that  place 

pKorosiTiON  III. — The  pitch  being  the  saTne,  the  ttrets  it  ij 
e  neloeitt/ 
bivcrval  opphcation  of  the  propoBition  by  reducing  tlie  fii'st 

o  the  same  standard  .         .         .     107 

0  be  horses"  power IO7 

d  by  the  greatest  number  of  horses'  power  ne- 
»  perform  the  work  of  assigned  trains  of  ma- 
chiuciy 
ScimtiBc  Hid  practical  values  of  horses'  power  illustrated     . 
Ueiagutier'B  measure ;   Smcalon's  measure ;  James  Watt's 


XVlll  CONTENTS. 

Alt. 

Number  of  spindles  of  cotton  t^ist  driven  by  one  bone       .     109  89 

Ditto  of  cotton  mule  yam 110  89 

Ditto  of  flax  yam Ill  89 

Comparison  of  different  estimates  of  the  force  of  moving 

powers         .         •         •         •         •         •  •         .112  89 

Immediate  force  of  men  and  horses  vrithout  deduction  for 

friction 113  89 

Performance  of  men  and  horses  by  machines       .         .         .114  90 

Practical  methods  of  calculating  this  force  .         .         .        .115  90 

Mechanical  power 116  90 

Exposition  of  this  power 116  90 

Demonstration  of  the  greatest  advantage  from  men  and 
horses  when  moving  with  half  the  velocity  they  would 
continue  at  work  were  the  effective  resistance  of  the  ma- 
chine nothing       117  90 

Demonstration  of  that  portion  of  the  mechanical  power  which 

is  efficient  in  impelling  the  machine         .         .         .         .117  91 

Practical  illustrations  of  the  strength  of  men  and  horses       .     118  91 

Strength  of  a  man  ascending  vertically  half  the  horizontal 

velocity 118  91 

Man  will  walk  d|  miles  an  hour  for  10  hours  a  day   .         .118  92 

His  maximum  of  effect  is  then  If  miles  an  hour,  or  2|  feet 

a  second 118  93 

Quantity  of  velocity  lost  in  friction  =  one-fifth  .  * .  maximum 

of  useful  effect  is  2  feet  per  second  .         .         .         .118  92 

Smeaton's  comparisons  of  animal  power  in  man  =  31*25  lbs. 
moving  with  a  velocity  of  2  feet  per  second,  or  to  ^  cubic 
foot  of  water  raised  2  feet  per  second ;  a  cubic  foot  of 

water  being  62|  lbs 118  92 

Bricklayers'  labourers  ascend  9  inches  per  second        .         .118  93 

Ascent  of  stairs,  &c 118  93 

Force  of  a  horse  equal  to  that  of  6  men,  according  to  esti- 
mates;  exertion  for  8  hours  about  2^  feet  per  second       .119  93 
Or  mechanical  power  of  a  horse  :=  187^  lbs.,  moving  with  a 
velocity  of  2|  feet  per  second;  or  to  3  cubic  feet  of 
water  raised  2^  feet  per  second ;  the  day's  work  being  8 

hours 119  93 

Equal  to  28,125  lbs.  raised  1  foot  per  minute  =:  a  mean  of 

Smeaton  and  Watt's  estimate 119  94 

French  and  American  dynamical  measure  of  power     .         .119  94 

Table  of  pitches  of  wheels  in  actual  use  in  mill-work,  exhibit- 


^^^^^^M 

■ 

^ 

^^P  tng  by  inspection,  the  kind  of  macliine,  power,  pitch  iti 

^^P  belies,  bnaidthoftectli  in  inches;  of  the  wheel,  its  number 

1 

■        of  teeth,  revohitions  per  minute,  and  diameter;   of  the 

1 

ptninn,  ita  nnmber  of  teeth,  revolutions  per  minute,  dii^ 

1 

■        meter;  the  breadth  proportionate  to  the  iiorscs'  power 

^^^J 

^^ft  Kod  present  velocity ;  present  velocity  in  feet  per  second, 

^^^fl 

^^Psnd  brcodtli  proportionnte  to  10  horses'  power  at  3  feet 

^^^H 

^^Bpcr  iecand;  that  is,  reducing  all  the  examples  to  the  some 

^^^H 

^B   deoomination 

120 

95 

^ESxpbnntian  of  the  tuble  of  wheels  in  actual  use  in  mill-work 

120 

95 

FfflMerratioiis  on  the  table  of  wheels  in  actual  uee  in  mill-work 

ISI 

96 

Rnle  1.  anil  esomple  for  tlic  construction  of  wheels  to  equal 

the  horses'  power  employed  in  the  macliinery . 

122 

97 

Description  of  six  (ablee  of  pitches 

122 

98 

Table  I.— The  velocity  of  the  pitch  line   being  3  feet  per 

•ecottd,  and  the  breadth  of  the  teeth  9  inches  . 

123 

99 

Table  II. — The  velocity  being  3  feet  per  second,  and  the 

breadth  of  the  teeth  doable  each  pitch    .... 

124 

99 

Table  III.— The  breiwith  of  the  teeth  8  inches,  and  velocity 

1 1  foct  per  second 

125 

100 

Table  IV. — The  velocity  being  constant,  1 1  feet  per  second. 

ond  the  breadth  constant 

126 

101 

Tobic  V. — At  a  velocity  of  3  feet  per  second,  the  breadth 

being  coustantly  B  inches 

127 

101 

Table  VI.— Velocity  at  3  feet,  the  breadth  being  double 

tJie  pitch 

128 

102 

Comporisou  of  these  tables 

128 

102 

Ealo  II. — For  a  pitch  of  3  inches,  with  a  velocity  of  3  feet 

per  second,  every  inch  of  breadth  being  valued  at  1 J  horses" 

power 

129 

102 

Braidtli  of  teeth  as  made  by  the  best  millwrights  now  seems 

to  be  about  twice  or  thrice  the  pitch       .... 

130 

103 

Bobcnon's  rule  for  the  teeth  of  wheels       .... 

131 

103 

JEIetnents  for  the  constructian  of  a  table  of  pitches  of  wheels 

132 

104 

^LZtble  of  pitches  of  wheels,  with  the  breadth  and  thicknesa  of 

^m 

^M 

^^Lmoring  Bt  tlie  pitch  line  at  the  rate  of  3  feet,  4  feet,  of  6 

^^ 

^^nfect,  and  of  8  feet  per  second 

133 

V 

1 

^^Kmu!  RAberton'e  tables  of  pitches 

134 

bs 

105                       ■ 

K 

1 

XX  CONTENTS. 

Art.  Pa^ 

Rule  by  Garmichael  for  calculating  the  proportionate  Btrength 

of  the  teeth  of  wheels 104         106 

Table  of  pitches  by  Carmichael,  founded  on  three  cases  of 

Roberton  8  and  three  cases  of  Buchanan's  tables       .         .134         106 
Explanation  of  the  table ;  pitch,  thickness,  breadth,  length 
and  strength  of  the  teeth ;  horses'  power  at  3,  6,  and  11 

feet  per  second 134         106 

Remarks  on  this  table 134         107 

Table  of  pitches  computed  from  the  rule  of  Garmichael,  with 
the  breadth  and  thickness  of  the  teeth,  and  the  correspond- 
ing strength  in  horses'  power 135         107 

Method  of  determining  from  first  principles  the  strength  pro- 
per for  teeth  of  wheels,  by  Tredgold  .  .  .  .136  108 
Rule  for  the  thickness  of  cast  iron  teeth  for  wheels. — Find 
the  number  of  horses  which  are  equivalent  to  the  power 
of  the  first  mover  of  the  train  of  machinery,  and  divide 
that  number  by  the  velocity,  in  feet  per  second,  of  the 
pitch  line  of  the  pinion  or  wheel ;  extract  the  square  root 
of  the  quotient,  and  three  fourths  of  this  root  will  be  the 
least  thickness  of  the  tooth  for  the  wheel  or  pinion,  in 

inches 137         109 

Rule  for  the  least  quantity  of  pitch  for  a  wheel  or  pinion 
with  teeth  of  cast  iron. — If  the  thickness  of  the  teeth  of 
the  pinion  be  intended  to  be  the  same  as  those  of  the  wheel, 
multiply  the  thickness  above  determined  by  2*1,  the  pro- 
duct will  be  the  pitch  required 138         109 

Example  to  illustrate  practically  the  foregoing  rule      .        «     139         110 

Of  the  thickness  of  wooden  teeth 140         110 

How  to  determine  the  breadth  of  cast  iron  teeth,  and  to  as- 
certain what  breadth  is  essential  to  strength     •         .         .141         111 
Rule  for  finding  the  breadth  of  cast  iron  teeth     .         .         .142         111 

Example  illustrative  of  the  rule 142         112 

Of  the  breadth  of  wooden  teeth ; — rule  and  example   .        .     143         112 
Of  the  strength  of  staves  for  trundles ; — ^rule  and  practical 

example 144         113 

Table  of  the  radii  of  wheels  from  10  to  300  teeth,  the  pitch 

being  2  inches 145         114 

Of  arranging  the  numbers  of  wheel-work    ....     146         115 
Rule  I.  For  the  number  of  teeth  in  one  pinion,  when  the 

wheels  drive  the  pinions 147         116 


CONTENTS.  XXI 

Art.  Page 

Rale  II.  For  the  some  when  the  pinions  drive  the  wheels    .     148         116 

Bole  III.  That  the  numher  of  teeth  in  a  wheel  should  not 
be  divisible  by  the  number  of  teeth  in  the  pinion  without 
a  remainder 149         116 

Rule  IV.  To  determine  the  exact  ratio  which  should  obtain 
between  the  teeth  of  the  wheel  and  that  of  the  pinion      .     150         117 

Of  calculating  tlie  numbers  for  wheel-work         .  .151         117 

Of  adapting  the  trains  of  machinery  to  produce  different  ve- 
locities at  the  working  points  151         118 

Of  the  position  of  the  first  mover  being  as  near  as  possible  to 

the  resistance 151         118 

Example  iUustrative  of  the  proportions  between  the  wheels 

Bud  "pinionsj  Bud  vice  versd 152         119 

Friction,  in  the  generality  of  combinations,  balances  two- 
thirds  of  the  power  appHed    153         120 

The  resistance  to  be  overcome  at  the  working  point  expends 

the  remaining  third 152         120 

Dr.  Jamieson's  demonstrations  of  these  facts       .         .         Pages  120-123 

Examples  illustrative  of  the  previous  demonstrations,  by  the 
same  author 154         123 

htcdcal  observations  with  regard  to  the  making  of  patterns 
for  cast  iron  wheels 155         124 

Role  for  making  the  length  of  the  teeth  equal  to  the  pitch, 

deducting  freedom 155         124 

Hafcton  on  clock-work,  his  rule 155         125 

Of  the  shrinking  of  metal  in  cooling 156        125 

IVoportions  that  have  been  found  to  answer  in  practice  fbr 
cast  iron  wheels 157         126 

Of  casting  wheels  in  parts,  and  oflcrwards  bolting  the  parts 
together 158         127 

Of  materials  for  patterns 159        128 


CHAPTER  V. 

Of  the  use  of  charts,  and  some  further  explanation  of  tlio  con- 
stmction  of  the  tables  of  pitches  of  wheel-work ;  showing 
the  horses'  power  to  which  the  teeth  of  wheels  of  certain 
pttdiesi  working  under  different  circumstances,  are  equal 

Qmotitiss  incraasiiig  or  decreasing  in  arithmetical  proportion 
jpcrMsing  in  geometrical  proportion    . 


160 

129 

161 

129 

162 

129 

163 

129 

164 

130 

165 

130 

166 

130 

167 

131 

168 

131 

XXll  CONTENTS. 

Art. 

Examples  illustrative  of  these  cases,  and  of  laying  down  such 

proportions  on  a  chart 

Method  of  tracing  on  a  chart  accelerating  motion 
Of  tracing  curves  from  proportions  on  such  tahles 
Mechanical  method  of  tracing  these  curves 
Various  uses  of  such  charts       .... 
Comparative  view  of  the  tahles  of  pitches  of  wheel- work 
Table  I. — Velocity  of  the  pitch  line  3  feet  per  second,  and 

breadth  of  the  teeth  9  inches,  and  value  of  strength  in 

horses'  power       .         .         .         .         .         .         .         .     168         131 

Table  II. — Velocity  3  feet  per  second,  breadth  of  the  teeth 

double  each  pitch,  and  value  of  strengtii  in  horses'  power  .     168         132 
Table  III. — Velocity   11   feet  per  second,  breadth  of  the 

teeth  8  inches,  and  value  of  strength  in  horses'  power      .     168         132 
Table  IV. — Velocity  1 1  feet  per  second,  the  breadth  double 

each  pitch,  and  value  of  strength  in  horses'  power   .         .168         133 
Table  V. — Velocity  3  feet  per  second,  breadth  of  the  teeth 

8  inches,  and  value  of  strength  in  horses'  power       .         .168         133 
Table  VI. — Velocity  3  feet  per  second,  breadth  double  each 

pitch,  and  value  of  strength  in  horses' power  . 
Reference  and  example  explanatory  of  Table  I.  . 
Reference  and  example  explanatory  of  Table  II. 
Reference  and  example  explanatory  of  Table  III. 
Reference  and  example  explanatory  of  Table  IV. 
Reference  and  example  explanatory  of  Table  V. 
Reference  and  example  explanatory  of  Table  VI. 
Explanation  of  the  chart,  in  regard  to  the  scales  representing 

the  line  of  pitches  and  horses'  power      .... 
Observations  on  the  intersection  of  the  curves     .         .      17( 


APPENDIX   A. 

Profesmtr  Willis  on  the  Teeth  of  Wheels, — Investigation  of 

the  curves  given  to  the  teeth  of  wheels  .         .         .180  139 

Section  I. — On  the  curves  adapted  to  practice  *.         .         .      —  139 

Instrument  to  illustrate  the  curve  and  furnish  a  practical  so- 
lution of  the  problem —  140 

Remarks  on  epicycloid  and  involute  curves  .         .         .       —  141 

•  The  numeriod  refiarence  of  180  applies  to  the  whole  of  this  Appendix,  and  it  is  not 
therafore  repeated  in  these  contents. 


168 

134 

169 

134 

170 

135 

171 

135 

172 

136 

173 

136 

174 

136 

175 

137 

179 

137 

CONTENTS. 


XXlll 


Art 


Proportions  that  obtain  in  epicycloidal  teeth 

Cbroflleirjr.— If  for  a  set  of  wheels  of  the  same  pitch,  a  con- 
stant describing  drclo  be  taken,  and  employed  to  trace 
those  portions  of  the  teeth  which  project  beyond  each 
pitch  line  by  rolling  on  the  exterior  circumference,  and 
those  which  lie  within  it  by  rolling  on  its  interior  circum- 
ference :  then  any  two  wheels  of  this  set  will  work  cor- 
rectly together 

Application  of  the  proportion  to  any  pair  of  wheels    . 

Method  of  settling  the  proper  diameter  to  be  given  to  the 
constant  describing  circle       ...... 

Application  of  the  proportion  and  corollary  to  racks    . 

Proofs  that  this  system  is  more  easy  of  practice  for  the  work- 
man than  the  old  one   ....... 

Of  a  form  of  increased  strength 

Analogous  to  the  wheels  of  watches  when  the  teeth  and 
pinion  leaves  are  of  a  saw-tooth  form     .... 

Section  II. — On  a  practical  approximation  to  the  true  form 
by  arcs  of  circles,  and  the  identity  of  Professor  Willis's 
method  with  that  of  Euler,  who  first  suggested  the  sub- 
stitution of  an  arc  of  the  circle  of  curvature  for  the  real 
curve  •...•.. 

Of  the  Odontagraph,  or  tooth  fashioner  *   . 

Tables  shewing  the  place  of  the  centres  upon  a  scale 

Centres  for  the  teeth  within  the  pitch  circle 

Centres  for  the  teeth  outside  the  pitch  circle 

Table  and  rule  for  finding  the  radius  of  the  wheel 

Chsometrical  construction  of  wheels  . 

Of  teeth  working  virith  trundles  or  radial  flanks 

On  cntters — and  the  method  of  obtaining  a  correct  form  of 
tooth  by  means  of  the  Odontagraph,  or  tooth  modelling 
mstnuuent  ......... 

Table  of  equidistant  values  for  cutters       .... 

Table  of  cutters 

SlCTiOH  III. — Theory  of  the  preceding  constructions  . 

Method  of  describing  teeth  consisting  of  a  single  arc  . 

Method  of  describing  teeth  consisting  of  two  arcs  of  circles 

Constniction  of  the  Odontagraph  or  tooth  modeller 

Example  shewing  how  this  instrument  is  connected  i^ith  the 
profioiis  demonstration 


Page 
142 


—  143 

—  143 

—  144 

—  145 

—  145 

—  14G 

—  147 


148 
150 
151 
151 
151 
152 
153 
154 


—  155 

—  156 

—  157 

—  157 

—  163 

—  166 

—  168 

—  168 


*  HoUMpiBlj  of  CSuring  Croa,  mikei  thb  iKUtmment 


XXIV  CONTENTS. 

Alt.  Pace 

Results  of  calculations  for  obtaining  a  principle  for  the  num- 
ber and  arrangement  of  the  wheds  selected    .        .        .     180        171 


ESSAY  II. 

On  the  shafts  of  mills,  gudgeons,  journals,  the  kinds  of  stress 
to  which  they  are  subject,  their  strength,  stiffiiess,  and  pro- 
portion         —  173 

CHAPTER  L 

Introductory  remarks  on  mill-work   .  .        »        .181  173 

Manner  in  which  this  essay  is  treated        ....     181  175 
Remarks  on  the  proportion  which  the  diameters  of  axles 

ought  to  have  to  the  stress  they  are  to  bear    .  .     182  170 

On  the  use  of  cast  iron  for  shafts 183  176 

CHAPTER  II. 

General  description  of  shafts —  178 

Distinction  between  shafts  and  spindles      .         .         .         .184  178 

Horizontal  and  Tcrtical  shafts 185  178 

Materiab  of  which  they  are  usually  made  .         .         .     186  178 

Wooden  shaft  laid  in  gudgeons 187  178 

Wooden  shaft  ^ith  cross^taikd  gudgeons  .  .188  179 

Of  hoops  on  shafts 189  179 

An  improved  method  of  fixing  gudgeons  invented  by  Robert 

Hughes 190  180 

Of  hollow  cast  iron  shafts 191  181 

Cast  iron  cylindrical  shaft,  which  npiy  be  variously  con- 
structed according  to  circumstances         ....     192  181 

Of  the  feathered  arrow  shaft 193  181 

Section  II. — Of  the  kinds  of  stress  to  which  shafts  are  sub- 
ject      —  182 

Of  lateral  stress  and  torsion 194  182 

Horiiontal  shafts  liable  to  lateral  stress  .         .194  182 

Upright  shafts  liable  to  torsion 194  183 

Stress  compouudoil  of  lateral  pressure        .  .        .194  183 

Roborton's  means  of  avoidiog  stress  and  friciioii  194  183 

Kxample  to  illiHtfate  tiieae  mmn  by  an  oreidMii 


CONTENTS.  XXV 

Art.  Page 

wheel,  shewing  that  the  stress  and  friction  of  the  gudgeon 
must  depend,  in  a  great  measure,  on  the  size  of  the  toothed 
wheel  attached  to  the  water-wheel,  and  to  the  situation  of 
the  pinion 195         183 

In  a  single  pair  of  wheels  of  whatever  form  or  construction, 
the  tendency  to  hreak  or  hend  the  shaft,  or  cause  friction, 
18  the  same  as  the  action  of  the  teeth  on  each  other         .     196        184 

In  the  case  of  an  intervening  wheel,  the  force  or  tendency 
to  break  the  shaft  depends  on  the  situation  of  such  inter- 
vening wheel 197        184 

A  wheel  placed  betwixt  two  others,  the  forces  being  equal 

and  opposite,  removes  the  strain  from  the  shaft  .198         185 

Of  the  place  on  the  shaft  on  which  the  wheels  are  fixed. 

Example  to  illustrate  this  case 199        186 

Of  the  strains  upon  journals,  and  placing  wheels  and  pinions, 
so  that  their  action  on  each  other  may  be  in  contrary  di- 
rections; to  avoid  the  strain  on  the  shaft  or  journals        .     200        186 

Of  lying  shafts  having  the  heaviest  shaft  on  the  lift  of  the 

wheels  to  take  off  the  friction  on  the  journals  .         .     201         187 

Roberton's  observations  on  the  foregoing  subjects        .        .    202        187 

By  increasing  the  size  both  of  wheels  and  pinions,  the  force, 
strain  and  firiction  on  the  shafts  and  journals  are  dimi- 
nished in  the  same  ratio 203        188 

ExAMPLB  I. — Illustrative  of  Roberton's  views     .         .  204         190 

ExAMPLB  II.— Illustrative  of  the  same  matter    .         .         .     204         190 

In  a  horse-gin,  where  the  pinion  is  driven  by  a  toothed  wheel 
on  the  gin,  the  friction,  or  strain  on  the  journals,  depends 
on  the  situation  of  the  horse  beam         ....     204         190 

In  a  water-wheel  turning  machinery,  the  strain  on  the  shaft 
and  teeth  are  the  same 205        190 

Example  to  illustrate  this 205         190 

Roberton's  judgment  of  Fenwick's  opinion  that  the  most 
perfect  machine  is  that  which  operates  with  the  fewest 
moving  parts 206         192 

When  wheels  differ  considerably  in  size,  the  gudgeon  next 
the  nnaller  wheel  will  have  to  sustain  the  greater  part  of 
tfaeatjesB 207        193 

PiMgnro  downward  on  one  gudgeon,  and  upward  on  another    208        193 

When  the  poaanre  at  the  gudgeon  is  wholly  in  a  lateral  di- 

209        193 


XXVI  CONTENTS. 

Art. 

Methods  of  computing  and  comparing  the  pressore  in  these 

different  cases 209         194 

Gasb  I. — The  power  and  resistance  being  at  opposite  sides 

of  the  shaft 209         194 

Gasb  II. — The  power  and  resistance  being  at  the  same  side 

of  the  shaft 209         194 

When  the  power  and  the  weight  are  oblique  in  respect  to 

one  another  .         . 210         195 

Methods  of  operation  by  the  resolution  of  forces         .         .210         196 


CHAPTER  III. 

Section  I. — Of  the  strength  of  gudgeons  where  the  stress  is 

produced  by  lateral  pressure  only —         197 

Of  the  size  and  strength  of  gudgeons         .         .         .         .211         197 

Pbop.  I. — Solid  cylinders  of  the  same  length  have  their  lateral 
strength  as  the  cube  of  their  diameters^  for^  in  general^  the 
lateral  strength  of  any  pieces  of  iron  or  timber^  whose  sec^ 
tions  are  similar  figures^  are  as  the  cubes  of  the  similar 
sides  of  the  sections 212         197 

Strength  of  gudgeon  limited  by  the  strain  it  will  bear,  with- 
out permanent  derangement  ..... 

Investigation  of  a  new  rule  for  the  strength  of  gudgeons     • 

Practical  rule  for  finding  the  diameters  of  gudgeons     . 

Comparison  of  the  rule  that  the  diameter  of  the  gudgeon 
should  be  equal  to  the  cube  root  supported  in  cwts. 

Sbction  II. — Of  gudgeons  of  water-wheels 

Introductory  remarks  on  water-wheels  of  various  weights, 
and  the  diameters  of  the  gudgeons  in  actual  use 

Description  of  the  first  table  of  gudgeons 

Notes  upon  empirical  rules 210 

Table  I. — Gudgeons  of  water-wheels  of  different  materials 

Observations  on  the  first  table  of  gudgeons 

Proof  that  the  cube  root  of  the  weight  in  cwts.  is  nearly 

equal  to  the  diameter  in  inches  of  the  gudgeon        .         .218         203 

Rule  for  finding  the  diameter  of  the  gudgeon  of  a  water- 
wheel. — The  cube  root  of  the  weight  of  a  water-wheel^  in 
hundredweightSy  is  nearly  equal  to  the  diameter  in  inches  if 
a  oast  iron  gudgeon  sufficiently  strong  to  support  such  wheel    219         203 

Example  illustrative  of  the  rule 219        204 


213 

198 

213 

198 

214 

199 

214 

199 

215 

200 

215 

200 

216 

201 

216 

201 

216 

202 

217 

202 

CONTENTS.  XXVU 

Art  Fife 

The  weights  of  OTenhot  or  bucket  water-wheels  will  be  to 
one  another  nearly  as  their  circumferences  or  diameters 

and  breadth 220         204 

Rnlc  for  the  diameter  of  the  gudgeons. — For  wooden  water^ 

wheds^  mtikiplif  the  diameter  in  feet  hy  the  width  also  in 

feeti  to  which  add  the  sgtiare  of  half  of  the  diameter.     The 

cube  root  of  the  sum  will  be  nearly  equal  to  the  diameter 

of  the  ^tufyeon  in  inches 220         204 

Example  illustratiTe  of  the  rule 220        204 

Explanation  of  Table  II.,  of  water-wheels  .  .  .221  205 
Table  II. — Gudgeons  of  water-wheels  .  .  .  .221  205 
Sbction  III. — Of  cast  iron  gudgeons  for  various  purposes   .      — -        205 

Introductory  remarks 222         205 

Explanation  of  the  table  of  cast  iron  gudgeons  .         .        .     223        206 

Table  of  cast  iron  gudgeons 223        206 

Use  of  the  table  shewn  by  Example  I.,  and  also  Example 

II 224        207 

Section  IV.-— Of  malleable  and  cast  iron  gudgeons  .  .  —  207 
I^ofessor  Robison  s  remarks  on  the  strength  of  cast  and 

wrought  iron 225         207 

Buchanan's  results  of  experiments  on  gudgeons  of  cast  and 

wrought  iron 225         208 

Method  of  finding  the  diameter  which  any  cast  iron  gudgeon 

should  have  to  sustain  any  given  pressure        .         .         .     226         208 
Note, — Tredgold  8  experiments  on  the  stiffness  of  cast  and 

malleable  iron 226         208 

Distinction  between  strength  and  stiffness  ....     226         208 
Example  shewing  the  method  of  finding  the  diameter  of  a 
wrought  iron  gudgeon,  having  given  the  lateral  pressure 
aod  the  diameter  of  the  cast  iron  gudgeon      .         .         .     226        209 
Explanation  of  a  table  of  cast  and  wrought  iron  gudgeons  .     226        209 
Table  of  cast  and  wrought  iron  gudgeons,  shewing  their 
lespective  diameters,  and  the  weights  they  can  sustain : 
the  diameters  of  the  cast  iron  swelling  from  1  inch  to  11 
inches,  and  the  wrought  from  1  to  9  inches,  and  the 
wei^ts  expressed  in  the  cubes  of  these  nmnbers     .         .     226         210 
Use  of  the  table  shewn  by  a  practical  example  .  .     227         211 

CHAPTER  IV. 
Bmtniom  Lf— Of  the  itrength  of  journals,  when  the  stress 


XXTUl  CONTENTS. 

An.  Plge 

arises  from  torgion  and  twisting  in  addition  to  lateral 

stress —        212 

Horses'  power  used  as  the  measure  for  the  strain  brought  on 

shafts  by  torsion  or  twisting 228        212 

Buchanan's  idea  that  wrought  iron  will  not  remst  torsion 

equal  to  cast  iron 228         212 

JVofe.— Definition  of  a  journal 228        212 

Section  II.— Of  proportioning,  journals  to  the  stress  which 
they  haTe  to  sustain —        213 

Illustrations  of  the  proportion  between  journals  and  the 
stress  tiiey  have  to  sustain 229        213 

In  all  cases  where  the  horses'  power  divided  by  the  revolu- 
tions per  minute  produces  the  same  quotient,  the  sta^ess  is 
the  same 229        213 

A  resistance  equal  to  50  horses'  power  making  50  revc^n- 
tions  per  minute,  produces  the  same  stress  as  10  horses' 
power  making  10  revolutions  per  minute        .        .         .     229         213 

Bules  for  calculating  the  strength  in  proportion  to  the  resist- 
ance        230         214 

ExAMPLS  I.F— When  the  horses'  power  and  the  revolutions 
per  minute  are  the  same  number 230        214 

Example  II.— To  find  the  diameter  of  the  journal  propor- 
tionate to  the  velocity 230        215 

Description  of  a  table  of  journals  proportionate  to  d  having 

420  as  a  multiplier 231         215 

Table  of  journals,  shewing,  1st,  the  horses'  power :  2dly, 
the  revolutions  of  the  journal  per  minute ;  3dly,  the  pro- 
duct of  the  povirer  divided  by  the  revolution  of  the 
journal ;  4thly,  the  proportionate  strain  on  the  journal ; 
and  lasdy,  the  diameter  of  the  journal  from  observation  .     231         216 

Observations  on  the  journals  of  fly-wheel  shafts  232        216 

And  on  secondary  shafts 232        217 

Multipliers  for  journals  of  steam-engine  fly-wheel  shafts  233        217 

Example  illustrative  of  the  table 233        217 

Note. — When  Buchanan  uses  the  word  journal,  he  supposes 
it  subject  to  torsion ;  where  there  is  lateral  pressure  only, 
and  no  torsion,  he  uses  the  word  gudgeon       .  .     233        217 

Rules  for  calculating  the  resistance  of  a  journal,  or  its  dia- 
meter as  regards  the  twisting  strain         ....     234         217 

Rule  I.— To  find  the  number  of  horses'  power  tiie  journal 
is  sufficient  to  resist 234        218 


CONTENTS.  XXIX 

Alt.  Page 

RuLB  11. — To  find  the  diameter  of  the  jotumal  in  inches      .     234        218 

Example  illasirative  of  the  rules 234        218 

General  rule  for  the  diameter  in  inches  of  the  journals        .     235         219 
Section  1X1^— When  the  diameter  of  a  joiumal  and  its  re- 
volutions per  minute  are  giyen,  to  find  the  horses'  power 

to  which  it  is  equal 236         219 

Rule  for  determining  this  case 236        219 

EzAUPLB   I.«»Shewing  the  horses'  power  to  which    the 

journal  is  equal 236         219 

Example  II. — ^When  the  journal  is  connected  with  heavy 

machinery 236         220 

Example  III. — The  same  journal  for  internal  work  of  the 

ordinary  kind 236        220 

CHAPTER  V. 

Section  I. — On  the  hodics  of  shafts  .        .        .        .      —        221 

Preliminary  remarks  on  the  distinction  hetweon  stiffness  and 
strength 237        221 

Shewing  that  the  limit  of  stiffness  is  flexure ;  and  the  limit 

of  strength  is  fracture 237        222 

The  laws  which  govern  stifihess,  and  those  which  determine 
strength 237        222 

Application  of  those  laws 237         222 

Of  lateral  stiffness,  and  lateral  strength       ....     238         222 

Proposition  II. — Any  beams  of  equal  length  have  their  lateral 
d^ness^  ^to  bear  a  load  at  any  point  in  the  length^  as  the 
breadth  and  cube  of  the  depths  and  have  their  latercd  strength, 
as  the  breadth  and  square  of  the  depth     ....     239         223 

Example  I. — Illustrative  of  the  comparative  stiffness  of  dif- 
ferent beams  or  shafts  239         223 

Example  II. — Illustrative  of  the  comparative  strength  of  dif- 
ferent beams        239         224 

Pbofosition  III. — Any  beams  of  different  lengths  have  their 
diffness  ^  bear  a  load  at  any  point  in  the  length']  directly 
as  the  breadth  and  the  cube  of  the  depth,  and  inversely  as 
Ike  cube  of  the  length,  and  have  their  strength  directly  as  the 
hreadthy  and  as  the  square  of  the  depth,  and  inversely  as 
tketatgik 240        224 

Ifate  upon  this  proposition   as  applied  to  practical  pur- 

240         224 


XXX  CONTENTS. 

Art. 

Example  I. — To  determine  the  comparatiYe  tft^ffnen    of 

beams  or  shafts  of  a  given  length  and  thickness       .        .     240        225 
ExAMPLB  II. — To  determine  the  comparatiye  sbreng^  of 

beams  or  shafts  of  a  given  length  and  thickness  .     240        225 

PaoposiTiON  IV. — SuppoHng  a  tube^  indefinitefy  t&tft,  to  be 
expanded  into  a  similar  tube  of  a  greater  diameter y  but  of 
equal  lengths,  the  quantity  of  matter  remaining  the  same^ 
the  STiFFNSss  will  be  inereasedy  in  the  ratio  of  the  square  of 
the  diameter,,  and  the  stbbnoth  in  the  ratio  of  the  diameter  241  226 
Example  I. — ^With  a  given  length  and  thickness  to  deter- 
mine the  comparative  stiffness  of  cylindrical  beams  or 

shafts 241         226 

Example  II.— To  determine  the  comparative  strength  of 

cylindrical  beams  of  a  given  length  and  thickness    .        .     241         227 
The  strength  of  shafts  is  increased  in  proportion  to  the  areas 

of  their  ends  and  diameters 242        227 

Professor  Robison's  remarks  on  cylindrical  beams  243        227 

Galileo's  observations  on  cylindrical  hollow  bodies       .         .     243        228 

Section  II.— Of  lateral  stress —        228 

Definitions  and  explanations  of  the  terms  stress  and  strain^ 

and  of  lateral  stress  in  particular 244        228 

Proposition  V. — The  stress  on  a  beam  arising  from  one 
weight  hung  upon  t^  is  proportional  to  the  rectangle  of  the 
parts  of  the  beam^  and  is  greatest  when  the  load  is  laid  on 

the  middle  of  the  beam 245        229 

Definition  of  the  rectangle  of  the  parts      ....     245         229 

Illustration  of  the  proposition 246         229 

Of  shafts  loaded  in  the  middle 247        229 

Of  the  load  united  in  the  centre  of  gravity         .         .         .     248         229 
Every  shaft  should  be  able  to  resist  the  strain  excited  at  that 

centre 249         230 

Shafb  subjected  to  lateral  stress  should  swell  in  the  middle .     250        230 
Shafts  of  the  form  of  a  cubical  or  semicubical  parabola        .     251         230 

Section  III. — Of  torsion —        231 

Proposition  VI. — In  general  the  strength  of  a  cylinder  or 
solid  axle  by  which  it  resists  being  wrenched  asunder  by 
twisting  is  as  the  cube  of  its  diameter      ....     252         231 

Of  hollow  axles 253        231 

Method  of  estimating  their  strength 253        231 

The  superiority  of  strength    of  hollow  tubes   over  solid 


CONTENTS.  XXXI 

Art.  Pace 

cylinders  is  mucli  greater  in  resisting  torsion  than  trans- 

Terse  or  lateral  stress 254        232 

Notes  and  illustrations  shewing  the  general  ratio  that  ohtoins 
between  the  strength  of  a  solid  cylinder,  and  that  of  a 
tube  containing  the  same  quantity  of  matter  .         .         .     254        232 

Professor  Bobison's  observations  on  the  adhesion  of  the 
fibres  in  wood,  and  the  molecules  of  metal  in  iron  shafts  .     254        233 

Of  the  excess  of  force  in  lateral  stress  and  twisting — or  when 

one  of  these  forces  exceeds  the  other     ....     255         233 

How  to  measure  the  resistance  of  a  lateral  stress         .         .     255        234 

One  hundredth  part  of  an  inch  the  quantity  of  flexure  that 
may  be  allowed  without  sensibly  affecting  the  regularity  of 
motion  in  a  shaft 255         234 

Method  of  calculating  the  stress  when  referred  to  the  middle 

of  a  cast  iron  shaft 255         234 

Circumstances  when  the  diameter  of  the  shaft  must  bo  de- 
termined by  the  rule  for  lateral  stress,  and  when  by  the 
rule  for  torsion 256         234 

Practical  illustrations,  shewing  that  the  bodies  of  shafts  need 

not  be  greater  than  tihe  journals 256        235 

Section  IV. — Application  of  the  foregoing  laws  practically, 

with  r^ard  to  the  proportions  of  shafts  ...       —         235 

Preliminary  observations  regarding  the  stress  upon  gudgeons 

or  journals 257         235 

Construction  of  water-wheels  without  shafts,  the  gudgeons 

being  fixed  to  the  arms  at  each  side  of  the  wheel    .         .258         236 

Practical  example  at  Cartside  mill,  in  the  Note   ...       —        237 

Cast  iron  shafts. — Ist.  Shaft  8  feet  long,  with  gudgeons  of  4 
inches,  is  weakest  in  the  middle ;  but  5  inches  in  the  mid- 
dle, it  would  be  as  strong  as  one  of  4  feet  long,  and  4 
inches  in  the  middle 259        237 

Of  the  stiffness  of  this  shaft  =:  that  of  one  6  feet  long,  and 

4  inches  throughout 259         237 

2nd.  When  the  point  of  greatest  lateral  pressure  is  2  feet 
from  one  end 260        237 

From  the  properties  of  the  lever,  the  gudgeon  next  the  point 

of  greatest  pressure  has  three-fourths  of  the  whole  to  sustain     260        238 

Bxamples  illostrative  of  these  cases,  and  of  the  strength  and 
•tiflbesB  when  the  shaft  is  reduced  to  a  given  section,  in 
order  to  enable  the  millwright  to  judge  how  the  shafts 
dMmld  amil  at  the  place  of  the  greatest  stress  .    260        238 


-.J 


XXXll  CONTENTS. 

Art.  P^ 

A  cylinder  is  stiffer  than  any  figure  that  can  be  inscribed 
within  it 261         239 

Rules  for  computing  the  diameters  of  different  forms  of  cast 

iron  shafts  to  resist  lateral  stress 261         239 

Istly.  If  the  stress  be  in  the  middle,  the  fourth  root  of  half 
the  stress  in  cwts.  multiplied  by  the  square  root  of  the 
length  in  feet,  is  equal  to  the  diameter  in  inches      .         .     261         239 

2dly.  If  a  cylindrical  shaft  has  no  other  lateral  weight  to 
sustain  but  its  own  weight,  multiply  the  cube  of  the  length 
by  *007,  and  the  square  root  of  ibis  product  is  the  diameter 
in  inches 262         239 

This  rule  enables  us  to  include  the  effect  of  the  weight  of 
the  shaft  itself.  Hence,  a  table  of  shafts  of  cast  iron  to 
resist  lateral  pressure ;  shewing,  Istly,  the  length  of  the 
shaft;  2dly,  its  diameter  in  inches  when  it  bears  only  its 
own  weight ;  ddly,  its  diameter  in  inches  when  the  stress  is 
equal  to  its  own  weight;  4thly,  its  diameter  in  indies  when 
the  stress  is  double  its  own  weight ;  5thly,  its  diameter  in 
inches  when  the  stress  is  three  times  its  own  wraght ;  and 
lastly,  when  the  stress  is  four  times  its  own  weight .        •     263        240 

Of  hollow  cylindrical  shafts  of  cast  iron^F— The  cube  of  the 
length  in  feet  multiplied  by  *009,  and  also  by  the  number 
of  times  the  weight  of  the  shaft  is  contained  in  the  stress, 
then  the  square  root  of  this  product  is  the  diameter  in 
mchee 264        240 

Table  of  hollow  shafts  of  cast  iron  to  resist  lateral  stress,  ex- 
hibiting, Istly,  the  length  from  4  to  16  feet;  2dly,  the  ex- 
terior and  interior  diameter  in  indies,  when  the  stress  is 
four  times  the  weight  of  the  diaft ;  ddly,  the  same  dimen- 
sions when  the  stress  is  six  times  the  weight  of  the  diaft; 
4thly«  when  the  stress  is  eight  times  the  weight  of  the 
shaft ;  and  StUy,  when  it  is  tm  timea  the  wei^t  of  the 
diaft 265         241 

Of  wrought  iron  dudb  to  rcdst  lateral  stress  .     266        241 

Of  wooden  sludb  of  oak  to  hare  the  same  stroigih  with  cmst 

iron  shafW  fite  udchce  square 267         242 

Of  the  comparatiTe  stiftMca  of  good  oak  sludb  as  compared 
to  thoeo  made  of  iron 268        242 

Example  lo  ilhislnile  the  fdatire  propottioii  of  an  oak  to  a 
c«sl  in^  shaft 269        242 

Of  the  slilbMt  of  iW  or  ysUew  fir  as  coHfttRa  to  casl  iron    270         843 


CONTENTS.  XXXlll 

Art.  Pnge 

Remarks  on  the  foregoing  data  and  examples  .        .271         243 

Phurtical  case,  showing  the  possibility  of  failure  when  excess 

of  strength  seemed  to  obtain  .....     272         243 

Hollow  cylindrical  shafts  equal  in  size  throughout  .  .273  243 
Of  shafts  subject  to  torsion,  especially  those  made  of  wood  274  243 
Practical  case  given  by  Buchanan,  wherein  the  shaft  was  only 

equal  in  strength  to  the  gudgeon 274         244 

Of  cross-tailed  gudgeons  of  wooden  shafts .  .         .274         244 

Of  cylindrical  shafts  of  cast  iron  to  resist  torsion  .     275         244 

Table  of  cylindrical  shafts  of  cast  iron  to  resist  torsion ;  com- 
prising, 1st,  their  diameter  in  inches ;  2dly,  the  number  of 
revolutions  from  5  to  50,  under  a  given  horses'  power      .     276        245 
Application  of  the  table  to  other  cases  in  which  the  shafts 

are  either  hollow  or  solid  cylinders         .        .         .        .277        246 
Bemark  upon  vertical  and  horizontal  shafts  loaded   with 

wheels 278         246 

Example  illustrating  the  foregoing  table      ....     278         246 
Of  a  shaft  of  cast  iron,  when  the  number  of  revolutions  is  20 
per  minute,  and  the  power  of  the  first  mover  equal  to  18 

horses 279        246 

Of  an  oak  abaft,  and  the  method  of  determining  its  diameter 

when  the  number  of  revolutions  is  40 ;  horses'  power  18  .    279        246 
When  fir  is  used  for  a  shaft,  its  diameter  should  be  2*06 

times  that  of  one  of  cast  iron  to  do  the  same  work  .         .281         247 
EzAJCPLB. — Power,  7  horses;  turns,  11|  per  minute;  to 
find  the  diameter  ==  5-8  inches,  being  of  cast  iron,  or  11 

inches  if  of  fir 281         247 

Allowance  should  be  made  in  fir  shafts  to  resist  torsion,  when 

the  abaft  has  to  sustain  both  lateral  strength  and  torsion   .     281         247 
EzAicPLXS — ^illustrative  of  the  sum  of  the  straining  forces  for 
acjlindrical  shaft  of  cast  iron  to  determine  their  diameter     281         247 

Ofthe  patterns  of  cast  iron  shafts 283        247 

Of  the  dimensions  of  shafts  subject  to  torsion      .                 .     284        248 
Ofthe  diameters  of  jonmals,  and  a  table  of  shafts  of  cast 
and  malleable  iron 284        249 

APPENDIX. 

CohMm  itrangth  of  different  metals 285        250 

itrangth  of  different  woods 285        251 

of  fiweigiierBy  philosophers,  and  engmeers  on  the 

c 


287 

254 

287 

254 

287 

254 

288 

255 

288 

259 

289 

259 

XXXIV  CONTENTS. 

Art. 

Strength   of  materials.  —  Table  of  MnsdienbroSk's  ex- 
periments on  the  strength  of  materials   •         •        .         •     285         252 
Emerson's  table  of  the  load  which  may  be  safely  suspended 

to  an  inch  square  of  various  materials     .         .         •         •     285         253 
Banks  takes  iron  to  be  4  times  as  strong  as  oak,  and  5| 

times  as  strong  as  deal  or  fir 286         253 

Results  of  yarious  authors  on  the  cohesiye  strength  of  ma- 
terials          

Strength  of  materials  in  resisting  compression      •         ^ 
Iron  more  liable  than  wood  to  accidental  imperfections 
Table  of  the  experiments  of  Brown,  Buffon,  Muschenbro^k, 
Perronet,  Rondelet,  Morveau,  Rennie,  Rumford,  Tredgold, 
Telford,  Sickingen,  from  the  Philosophical  Magazine 
Remarks  upon  this  table,  the  most  extensiye  of  its  kind 

Experiments  on  alloys  of  the  metals 289 

Copper  and  tin,  by  Muschenbroek ;  gun  metal  and  brass,  by 
Rennie ;  English  tin  and  lead,  Muschenbroek ;  Banca  tin 
and  antimony,  by  the  same ;  Banca  tin  and  bismuth,  by 
the  same  ;  Banca  tin  and  Indian  sine,  by  the  same ;  Eng- 
lish tin  and  zinc,  by  the  same ;  English  tin  and  antimony, 
by  the  same ;  Dutch  lead  and  bismuth,  by  the  same  •     289        260 

Observations  on  the  composition  of  these  alloys  .         .        .     289        261 
Authorities  for  the  cohesive  force  of  woods  of  various  kinds, 

are  Tredgold,  Muschenbroek,  and  Barlow       .         .         .     289        261 

ESSAY   III. 

On  the  construction  and  durability  of  the  longitudinal  con- 
nexions of  shafts,  denominated  couplings         ...       —        262 

Preface.— Different  methods  employed  in  coupling  shafts     .       —        262 

Remarks  on  the  fallacies  of  some  eminent  men  in  applica- 
tions of  favourite  theories —         264 

Dr.  Robison's  strictures  on  the  blimders  of  practical  men  who 

disregard  entirely  scientific  knowledge    .         .         .         .       -i-        265 

CHAPTER  I. 

On  the  longitudinal  connexions  of  shafts,  denominated  cou- 
plings           290  266 

Class  I. — Of  couplings  with  two  bearings  .        .        .  291  266 

Coupling  I.^^f  the  square  coupling        ....  292  267 

Of  the  oblong  coupling 292  ^67 


CONTENTS.  XXXV 

Art.  Page 

Remarks  on  these  coupliDgs,  and  on  the  imperfection  called 

Ali/i 293  268 

Coupling  of  rollers  as  mules 293  268 

Couplings  with  donhle  hearings,  how  made  .  .294  268 
Coupling  II. — Of  the  round  coupling  ....  295  268 
Oheervations  on  the  effects  of  round  coupling  .  .  296  269 
CovPLiNe  III. — Of  clutches  or  glands,  having  douhle  hear- 
ings      297  269 

OhservationB  on  glands  or  couplings  for  douhle  hearings  298  269 

Methods  of  adjusting  the  arms  and  points  of  glands               .  299  270 

Coupling  IV. — Of  the  horing  mill  clutch ;  first  construction  300  270 
Oheervations  relating  to  the  application  of  this  coupling  to 

slow  work 301  271 

Coupling  V. — Second  construction  of  the  horing  mill  clutch  303  271 
Observatioiis    showing  this    to    he  a  stronger  and  hotter 

clutch 304,305  272 

Coupling  VI. — Having  two  round  plates  that  serve  to  engage 

theahafls 306  272 

Ohservations  to  show  the  durahility  of  this  clutch,  or  species 

ofglands 307  272 

Naie^  upon  making  the  circular  heads  toothed     ...  —  272 

Coupling  VII.— Boulton  and  Watt's  coupling  link  •  .  308  273 
Ohservadons  showing  the  durahility  of  this  coupling,  and 

that  the  axes  move  without  twisting       ....  309  273 

Of  the  length  of  the  crank 309  273 

Coupling  VIII. — For  conveying  motion  to  a  fly  wheel  .  310  273 
Observations  on  the  durahility  of  this  coupling,  and  its  ap- 

plicalnlity  to  thrashing  mills 311  274 

OufKBAL  Obsbbvations. — Firstly,  on  friction,  and  couplings 

with  one  bearing 312  274 

Secondly,  when  heavy  drums  or  wheels  are  placed  near  the 

ends  of  the  shaAs,  two  hearings  must  he  used .         .         .313  275 

CHAPTER  II. 

CSlajm  IIh— Of  couplings  having  one  hearing        ...  —  276 

Smxion  L— Hook's  universal  joint 314  276 

Rojperiy  of  lihe  universal  joint  for  communicating  angular 

aotkn 315  276 

lb  i1iwilwntig,n  of  the  universal  joint                               .316  277 

e  2 


XXXVl  CONTENTS. 

Art.  P«ce 

Couplings  described  in  Chapter  I.  may  be  converted  into 

couplings  haying  one  bearing  .         .         .         .         .317         277 

Coupling  IX. — The  square  coupling         .         .         .         .318         277 
Observations  showing  the  efficiency  of  this  coupling  in  con- 
veying motion  through  a  great  length  of  shafts         .         .319         277 
This  coupling  liable  to  lifting  or  straining    .         .         .         .319         278 

Mules,  having  only  one  bearing 320         278 

Accuracy  required  in  mtde  and  throstle  rollers      .         .         .     320         278 
Notes, — Samuel  Crompton,  the  inventor  of  the  mule ;  Ark- 
wright's  patent  for  preparing  cotton  by  machinery ;  inven- 
tion of  the  throstle —         278 

Coupling  X.— Of  the  round  coupling,  having  only  one 

bearing 321         279 

Observations  on  round  coupling 322         279 

Coupling  XI. — With  a  scarfed  joint  ....     323         279 

Observations    on  this  contrivance,  showing  that  it  has  all 

the  defects  of  a  solid  shaft 324         279 

Coupling  XII.— A  variety  of  Coupling  XI.       .        .        .     325         280 
Coupling  XIII. — Another  modification   of  Coupling  XL, 
having  one  shaft  firmly  fixed  to  the  other  by  flanches  and 

bolts 326         280 

Coupling  XIV. — Has  the  bearing  and  the  joint  of  the 
coupling  at  the  same  parts  of  the  shaft,  the  ends  of  which 

are  quadrants 327         280 

Observations  showing  this  coupling    to  be  attended  with  ' 

trouble  and  expense 328         280 

Coupling  XV.  consisting  of  three  distinct  parts  which  join 

into  one  another ;  is  well  explained  in  the  plate      .         .     329         281 
Observations  on  the  first  cost'  of  this  coupling,  or  universal 

joint,  showing  its  excellency 330         281 

Coupling   XVI.    a    contrivance   executed  in   Buchanan's 

time  at  Manchester 331         282 

Observations  showing  that  the  principles  of  this  coupling 

agree  with  those  of  the  square  coupling  ....     332         282 

Advantages  of  this  coupling 333         282 

Coupling  XVII.  as  used  extensively  at  Glasgow  .  .  334  282 
Observations  on  the  advantages  of  this  kind  of  coupling  .  335  283 
Section  II. — Of  the  couplings  of  upright  shafts  ...  —  283 
Square  coupling  applied  to  these  shafts  ....  336  283 
Coupling  XVIII.  is  described  by  the  diagram  in  the  plate    337        283 


CONTENTS.  XXXVll 

* 

Art.  Pa^ie 

Coupling  XIX.-- This  also  is  well  described  by  the  diagram 

in  the  plate 338         284 

Obserrations  on  this  kind  of  coupling,  as  for  light  work, 
especially  flour  mills,  and  connecting  the  feeder  with  the 
stone-spindle 339         284 

Coupling  XX. — This  also  is  explained  by  the  diagram  in 
the  plate 340        284 

Obsenrations  showing  this  a  good  and  simple  mode  of 
coupling  upright  shafts 341         284 

CHAPTER   III. 

GsNERAL  Observations —        285 

The  laiger  the  parts  of  couplings  can  be  made,  so  much  the 
better 342         285 

The  further  the  point  of  stress  is  from  the  axis,  the  couplings 

will  be  more  durable     .......     343         285 

Ezemplificatioiu>f  this  in  the  handspike,  capstan  bar,  or  simi- 
UirleTer 343         285 

Practical  illustrations  of  the  correctness  of  these  observations 
in  point  of  durability 344         285 

In  along  line  of  shafts,  the  couplings,  where  there  is  only  one 
bearing,  should  be  so  arranged  that  the  unsupported  end 
of  the  shaft  should  be  as  far  as  possible  from  the  part 
subject  to  lateral  pressure ;  illustrated  by  diagram    .         .     345         28 G 

Oiling  of  couplings  renders  them  more  durable    .         .         .     346         286 

Fly  wheel  often  used  in  a  long  line  of  couplings  .        .     347         286 

Table  representing  the  dimensions,  stress,  and  durability  of 
ooapHngs,  in  nine  cases  of  facts  ;  combining  the  resistance 
in  horses'  power,  the  revolutions  per  minute,  the  number 
of  years'  work,  the  side  of  the  square  in  inches,  the  length 
of  the  box,  and  the  comparative  stress    ....     348         287 

Obsbbvations. — I.  On  the  durability  of  these  couplings      .     349         288 

II.  CSrcmnstances  affecting  their  durability         .         .         .350         288 

III.  Standard  of  power  and  strength  .351  288 

lY.  Vdod^  affects  the  durability 352  280 

V.  daasification  of  couplings  with  respect  to  durability        .     353  289 
BvpFunmrTABT  Obsbbvations. — I.  On  the  durability  of 

eoiiidings 354  289 

IL  Thar  durability  depends  mainly  on  accuracy  of  work- 

nmhip 355  289 

nL  Jhmnrj  of  dmvlnlity  deduced  from  construction    .  356  289 


XXXviii  CONTENTS, 

An. 

ESSAY  IV. 

On  the  methoils  of  disengaging  and  re-^engaging  machinery 

while  in  motion -—         291 

Intboduction. — Plan  followed  in  this  Essay      ...      —        291 

Methods  of  disengaging  shafts 357        292 

Of  the  vU  inertia  of  matter 358         292 

Note. — Dr.  Young's  definition  of  this  term  ...      —        292 

Inertia  simply  indicates  that  matter  never  changes  its  state, 

unless  there  be  a  change  in  the  power  or  powers  acting 

upon  it 359         293 

Note. — Newton's  definition  of  vis  inertia   ....       —        293 
lUustration  of  the  strength  of  machinery  by  throwing  a  wheel 

into  gear 360         294 

Division   of    the  subject — engaging  and  disengaging  ma- 

chineiy — into  two  parts 361         294 

I.  Of  methods  used  when  motion  is  communicated  by  means 

of  bands,  belts  or  chains.     II .  Of  methods  when  motion 

is  communicated  by  means  of  wheel-work      .         .         .     361         294 
Method  I. — The  sliding  pulley,  Fig.  1,  an  old  contrivance; 

description  of ;  engraving  and  plate         ....     362         294 
Observations  relating  to  the  application  of  this  contrivance 

to  cotton  carding  machines 363         295 

Method  II. — The  bayonet,  Fig.  2 ;  full  description  of  this 

invention 364         295 

Note, — Pointing  out  some  of  the  many  ways  of  making  and 

appKing  this  contrivance —        296 

Observations  showing  the  superiority  of  the  bayonet  to  the 

sliding  pulley 365         296 

Method  III. — Of  the  lock  pulley;  description  of  this  in- 
vention, and  how  it  unlocks  and  disengages  the  pulley      .     366         297 
Observations  showing    that  this  invention  has  never  been 

much  adopted 367         297 

Method  IV. — The  fast  and  loose  pulley;  description  of  this 

invention 368         297 

Observations  on  the  belts  used  in  machinery  running  over 

pulleys 369         298 

The  fast  and  loose  pulley  remarkable  for  simplicity  .  •  370  298 
Its  application  in  cotton  mills  is  now  general  .  .  .  370  298 
Method  V. — ^Description  of  this  method— or  sack  tackle  371  298 
Observatioiit  on  the  sack-tackle 372        299 


CONTENTS.  XXXIX 

Art. 


Sbctior  II.— Of  the  metliods  used  when  motion  is  oonyeyed 

hy  means  of  wheel-work 373  299 

Of  throwing  a  wheel  into  gear 373  299 

MiTHOD  Vh — ^DisengBging  and  re-engagiDg  wheels  by  means 

ofbridgea 374  299 

Obaervationa  on  this  mode  of  disengaging  wheels  375  300 
MiTHOD  YII. — ^Wheel  with  sliding  clutch,  which  may  be  en- 
gaged or  disengaged  at  pleasure,  described  and  illustrated  376  300 

Obaervationa  on  its  utility 377  301 

Mbthob  VIIL— The  friction  clutch           ....  378  301 
Observations. — ^Applicability    of   this  contrivance    to    the 

largest  machinery 379  302 

MsTHOD  IX. — The  friction  cones 380  302 

Observations. — ^Application  of  these  cones  to  sack-tackle  381  303 

MsTHOD  X.^- Wheels  acting  by  friction      ....  382  303 
Obeervations. — Used  with  good    effect    in    machinery  for 

raismgcoal 383  303 

Mbthod  XI. — Tackle  for  raising  sacks  in  a  brewhouse  384  304 
Observations  on  the  ingenuity  of  this  invention  .        .         .  385  304 
IfXTHOD  XII. — Self  disengaging  coupling— figure  represent- 
ing the  coupling  as  disengaged 386  304 

il^.^-IUustration  of  this  method  founded  on  Coulomb's  ex- 
periments      386  304 

Observations.— This    coupling  very    useful   when    turning 

lathes  are  driven  by  wheel-work 387  305 

Jfoie.'-^On  the  comparative  merits  of  wheels  387  305 

ESSAY  V. 

On  mechanism  for  equalizing  the  motion  of  mills,  denomi- 
nated lifi  tenters,  engine  governors,  and  water-wheel  go- 
ynmon —        307 

IlTTBODUcnoN. — Showing  that  this  essay  relates  to  machinery 
not  leas  curious  in  its  construction  than  useful  in  practice.' 

Of  the  steam-engine  governor,  and  throttle-valve  described  . 

JVoCstd— Descriptive  of  the  throttle- valve    .... 

SaonON  h — The  steam-engine  governor — its  particular  cou- 
■tnietioii  ........ 

Opention  d  the  governor  and  throttle-valve 

Fopokr  dewription  of  the  whole  apparatus 

▼HmUmm  of  (he  pendulum 


— 

307 

388 

308 

388 

308 

389 

308 

390 

309 

391 

309 

392 

310 

xl  CONTENTS. 

Art. 

Of  the  lengths  of  pendulums  and  oscillations  in  one  minute 

of  time 393         310 

Example  showing  how  to  find  the  lengths  of  pendulums      .  394         310 

Section  I. — Of  the  windmill  lift-tenter     ....  395         311 

First  construction  of  lift-tenters  for  windmiUs     .         .         .  396         311 

Second  construction  of  lift^tenters,  drawn  at  Liyerpool         .  397         312 
Section  III. — Of  governors  applied  to  water-wheels,  and 

made  on  various  constructions 398         312 

First  construction  of  the  water-wheel  governor  .         .         .  399         313 
Method  of  lifting  a  wheel  out  of  gear  when  a  mill  is 

stopped .         .  400        315 

Second  construction  of  a  water-wheel  governor  .         .         .401  315 

Third  construction  of  a  water-wheel  governor     .         .         .  402         315 
Ohservation  shewing  that  wheel-work  is  preferahle  to  hands 

and  pulleys 402        316 

Fourth  construction  of  the  water- wheel  governor         .         .  403         316 

Fifth  construction  of  the  water-wheel  governor  .         .         .  404        317 

Appendix  on  the  velocity  of  water-wheels         .         .        .  —        318 
Difficulty  of  finding  a  law  of  universal  application  for  giving 

different  degrees  of  velocity  to  water-mills      .         .         .  405         318 

Experiments  on  the  Rothesay  Mills,  hy  Buchanan       .         .  406         319 
Methods  of  ascertaining  the  proportional  quantities  of  water 

used  hy  the  old  mill 407         320 

Smeaton  and  Buchanan's  experiments  compared           .         .  408         321 
Buchanan's  experiments  are  consistent  with  the  experiments 

of  Smeaton 409         321 

Roherton's  observations  on  Buchanan's  and  Banks's  experi- 
ments             410         322 

Illustration  of  these  remarks 410         323 

Roherton's  remarks  on  overshot  wheels,  and  comparative 

value  of  work  and  water  used  in  performing  that  work    .  411         325 
On  overshot  wheels. — The  two  principal  elements  to   be 

considered  in  the  theory  of  wheels         .         .         .         .412         326 
On  the  proportion  of  the  radius  of  the  wheel  to  the  height 

of  the  fall 412         326 

Demonstration  of  this  proportion 412         326 

Important  practical  rule  or  maxim  deduced  therefrom  412        327 
Method  of  finding  the  effective  height  of  the  fall  and  radius 

of  the  wheel 412         327 

Of  the  velocity  of  the  circumference  of  the  wheel  to  pro- 
duce a  maximum  of  dTect     413        328 


CONTENTS.  xli 

Art.  Page 

Friction  eqaal  to  two  thirds  the  moving  power — the  velocity 
of  the  drcumference  of  an  overshot  wheels  in  feet  per 
second,  should  he  2*67  times  tiie  square  root  of  the  whole 

height  of  the  M  in  feet 413        329 

To  determine  the  part  of  the  fall  which  wiU  give  the  water 

the  same  velocity  as  the  wheel 413        329 

Comparison    of   these    results  with    the    experiments   of 

Smeaton 413        329 

On  compntiBg  the  power  of  overshot  water-wheels  .  •  414  330 
Equation  for  the  effective  force  of  the  water       .        .         .414        330 

Ditto  for  the  mechanical  power 414        330 

When  the  wheel  is  supplied  at  the  summit,  the  power  is 

equal  to  half  the  weight  of  water  supplied  to  the  wheel  .     414        330 
Comparative  power  of  overshot  and  hreast  wheels       .        .414        331 
Two  points  of  view  under  which  tiie  power  of  a  water- 
wheel  must  he  considered 414        331 

Method  of  estimating  the  horses'  power  which  any  water- 
wheel  may  have 415        331 

Examples  shewing  the  horses'  power  in  overshot  and  hreast 

wheels 415        332 

And  also  the  effective  force  when  the  water  flows  on  either 
at  the  summit  or  the  level  of  the  axis     ....    416        332 

Of  the  power  of  hreast  wheels 417        332 

Smeaton's  comparison  of  the  mechanical  power  of  an  imder- 

shot  and  overshot  wheel 417        332 

ESSAY  VI. 

On  changing  the  velocity  of  machinery  while  in  motion  .  —  334 
Intboduction. — Division  of  machinery  into  mill-work  and 

smaller  machinery. — The  mechanism  descrihed  in  this 

Essay  helongs  to  the  latter  class -*        334 

SicnON  I. — Of  turning  lathe  friction,  and  of  helts  of  the 

same  length  working  on  opposite  pulleys  .  .  .419  335 
Ohaervations  on  the  series  of  truncated  cones,  &c.,  in  this 

contrivance 420        335 

Of  alteniate  cones,  or  one  cone  giving  motion  to  another  .  421  336 
Ofaaermtion  on  this  piece  of  mechanism  ....  422  336 
Allentioii  of  velocity  hy  wheels  moving  one  another  hy 

findion 423        336 

Ohiii  f alkim  on  the  pecoliar  use  of  these  wheels  or  cones   .    424        337 


xUi  CONTENTS. 

An.  Pv 
Sbotion  n. — Moles,  well  adiqrted  for  apiiudng  all  kind*  of 

«/» 4S5  337 

James  Crompton,  the  inyentor  of  die  mule                         .  *S5  337 

Williun  Kelly  of  Lanark's  patent 425  338 

Velocity  of  spindles  called  doable  speed     .         .         ■         •  4S5  338 
Contrivances  to  show  the  progress  of  improTemcnt  in  this 
species  of  machinejj,  and  hence  the  first  constnictioii  for 

donble  speed *»6  338 

Obscnalions  on  this  conatnction **7  338 

Beeond  construction  for  donble  speed          .         .         .        ■  438  889 

ObsorvtitioiiB  on  this  construction *^9  840 

Third  construction  for  douUe  speed *30  340 

Obsemtioiis  on  this  ooMtruction ^1  .      ^1 

ESSAY  VII. 

On  the  fisming  of  mill-woik ^  34i 

pRBPACB— relating  to  the  moving  parts  of  machinery  .  —  342 

SxoTiON  I.— Pecnliaritiea  of  framing  of  mill-work       .         .  433  342 

Causes  which  subject  it  to  speedy  decay     ....  433  343 
Qualides  which  miU-work  shonld  poeseea  to  make  it  dnraUe, 

alrmgA,  sfj^uss,  aitd  mJuiHf 433  343 

Construction  should  he  such  that  any  particnlar  part  may  be 

repaired  or  renewed  with  the  least  possble  derangement 

to  the  other  parts 434  343 

Of  repkeing  dufts 435  343 

Fiiotion  diminished  by  the  elastic  powv  id  madunety  436  344 

Sktiok  II. — Of  the  bearings  of  shafts     ....  437  344 
Of  steps,  bushes,  bnaata,  pillow  hloeka,  plumber  hloc^  pe- 

dsatals 437  844 

The  subslancee  nasd  fo  pilktwa 438  844 

iB^woTtueiits  by  Mr.  Hnrray,  of  Leeds    ....  438  345 
Mstkods  adopts^  by  Oe  Shd&eM  griadcn  in  the  oso  of 

tborotiat 438  845 

A'stw  upon  nesal  and  wooden  pillows       ....  —  345 

Of  the  teims  of  Mi^w,  anj  of  npngbt  Aafb — first  mode     .  439  346 

Seeondmode 440  346 

Om««  in  wUn^  tbe  1^101  and  step  do  Mt  awver  w«U           .  44t  347 

TW  <|y.fi>nn«d  |«i«t 44S  347 

BiMwab's  Mod*  t«f  nuMSf  fawit*  in  a  iaid  Vr  wiaai  of  a 

.  443  347 


CONTENTS.  Xllii 

Art.  Page 

Breasts  and  bnshes 444  347 

Fnnnels  and  spindles 445  847 

Sktion  III. — Of  wooden  framing —  848 

Headstock  framing 446  848 

Fimming  for  lying  shafts 447  848 

Methods  of  framing  the  parts,  and  suspending  the  shafts  from 

a  ceiling 448  849 

Of  the  framing  of  upright  shafts— of  screws  and  wedges     .  449  849 
Of  the  framing  of  upright  and  lying  shafts,  connected  by  be- 

Telled  wheels 450  849 

Bridge  supported  by  doves 451  849 

Respecting  the  decay  of  timber,  and  the  means  of  prevent- 
ing that  decay 452  849 

Sbction  IV. — Of  cast  iron  framing 458  850 

The  resistance  of  cast  iron  to  compression          .         .         .  458  850 

Remarks  on  the  uniform  strength  of  cast  iron     .                 .  454  851 

Of  the  strength  of  cast  iron  beams 455  851 

Illustrations  of  sections  of  cast  iron  beams          .         .         .  456  851 

Limit  of  atrength  \»  fracture^  oi  stiffness  \s  flexure        .  456  851 

Of  feathered  cast  iron  framing 457  852 

Methods  of  making  cast  iron  framing  to  imitate  wooden 

framing 457  852 

Of  wood  and  iron  bridges  for  sustaining  shafts    .         .         .458  Z5Z 

Hollow  cylinder  applicable  in  many  cases  ....  459  ^5Z 

Headstock  of  cast  iron 460  Z5Z 

Various  modes  of  suspending  shafts  from  ceilings        .         .461  S5Z 

Bleaching  machine  called  squeezers    .         .         .         .         .  462  853 

Description  of  these  squeezers 462  253 

ESSAY  VIII. 

Ctoometrical  and  practical  method  for  finding  the  centres  of 
gravity  of  mill- wheels;  illustrated  by  examples,  in  which 
two,  three,  and  four  wheels  compose  the  system  upon  one 

and  the  same  shaft —  354 

Method  of  finding  the  centre  of  gravity  of  two  bodies         .  463  354 

Geometrical  construction  of  this  method — showing  that  the 

centre  of  gravity  is  known  in  terms  of  the  masses  .         .  468  854 

Flncticd  BxHen^^MuUipli/ either  body  by  the  whde  distance  be- 
iweem  Anreemiru:  divide  the  product  by  the  sum  of  the 
1/  Cl#  fmHetU  witt  be  the  distance  from  the  centre  of 


Xliv  CONTENTS. 

Art. 

gravity  of  that  body  opposite  to  the  one  by  which  the  whole 

distance  is  multiplied 464        356 

Example  to  illustrate  the  rule 464        356 

Analytical  investigation  when  the  weight  of  the  shaft  is  in- 
cluded       465         356 

Practical  Rule. — To  twice  the  weight  of  either  hody^  add  the 
whole  weight  of  the  lever  or  connecting  har^  and  mtdtiplg 
the  sum  by  the  central  distance  ;  then  divide  the  product  by 
twice  the  mass  compounded  of  the  bodies  and  the  bar,  and 
the  quotient  wiU  be  the  distance  of  the  centre  of  gravity 
from  that  body  opposite  to  the  one  whose  double  is  employed 

in  the  first  step  of  the  operation 466         357 

Example  I. — To  iUustrate  the  rule,  and  shew  the  positions 
of  the  wheels  relating  to  the  common  centre  of  gravity  of 

the  shaft 466         357 

ExAMPLB  lid — Bodies  of  unequal  weight  at  the  ends  of  the 

shafts,  to  find  common  centre  of  gravity  .         .         .     466        357 

Of  three  bodies  connected  by  an  inflexible  bar  .  .  .  467  358 
Demonstration  of  their  common  centre  of  gravity  .  .  467  358 
Practical  rule  derived  irom  the  demonstration  .  .  .  468  358 
Example  showing  how  to  find  the  common  centre  of  gravity 

of  three  bodies 468         358 

Dr.  Jamieson's  method  of  verifying  these  results  in  his  Me- 
chanics for  Practical  Men 469         359 

The  cases  of  utility  consistent  with  this  theorem         .        .    470         360 
Example  to  illustrate  the  position  of  an  intermediate  wheel, 
or  that  the  whole  weight  may  be  on  the  middle  of  the 

shaft 470         360 

When  the  distance  is  known  or  limited  by  situation,  and  the 
common  centre  of  gravity  must  fiedl  at  the  middle  of  that 

system 471         361 

Example  to  illustrate  this  case,  there  being  3  wheels  of  im- 
equal  weights  to  be  supported  by  a  girder  placed  at  the 
common  centre  of  gravity  of  the  system         .        .         .     472         362 
Verification  of  the  result  now  obtained     ....     473         363 
When  the  weight  of  the  axle  of  the  wheels  is  given  .         .     474        363 
Example  of  3  wheels  of  imequal  weights  on  a  shaft,  and  it 
is  required  to  find  the  common  centre  of  gravity  for  a 

support 474         364 

Of  the  centre  of  gravity  of  four  or  more  bodies  situate  in  the 

same  right  line     •        ...  •        •        .        •    475        365 


Dcmonstntioa  of  lliis  ciue,  wliicb  is  but  an  cxtcasion  of  tbe 
fonner         .........     475 

—MuUiplg  llie  maffnitttdt  or  density  of  tach  body  by  Us 

Tttlive  diitance  from  the  htginning  of  the  tyttem,  and 

'    divide  the  turn  of  ike  products  by  the.  turn  of  lie  bodies  for 

lie  diilajiee  of  the  centre  of  gravity  sought       .         .         .     476 
EiAMPLK  I. — Of  four  bodies  on  the  some  shaft,  and  it  is 

required  to  find  their  coraraon  centre  of  support      .         .     476 
Geometrical  construutioD,  to  shew  the  example  or  similar 

examples  may  be  worked  mechanically  .         .         .     477 

Example   II. — Sliewing  the  exact  distance  of  each  of  four 

nbecls  from  the  common  centre  of  grarity      .         .         .     477 

Of  the  centres  of  gravity  of  cones,  and  of  a  conic  frustnm  .     479 

Rule. — To  the  sum  of  the  squares  of  the  rmlii  of  tite  two  ends 

add  their  prodari,  then  multiply  the  sum  liy  4,  and  reserve  the 

remit  for  a  dicisor. — To  three  tivitt  tite  tquare  of  the  radius 

^^     ef  the  greater  emf,  add  the  square  of  the  radius  of  the  less 

^^L    end,  together  vith  twice  the  product  of  the  radii,  and  mul- 

^^P    liply  the  sum  by  the  height  of  the  frustum  for  a  dividend. 

^^B     — Then,  divide  the  dividend  by  the  reserved  divisor,  and 

the  piotienl  will  express  the  distance  betieeen  the  centre  of 

magnitude  of  the  lees  end,  and  the  centre  of  gravity  of  the 

frustum 476 

Of  die  centre  of  gmvity  of  the  Burface  of  a  cylinder  .         .     478 
Of  the  centre  of  gravity  of  a  circular  arc  ...     478 

Of  tbe  centre  of  gravity  of  a  parabola,  Bemiparabolo,  and 
paiabobc  conoid   ........     478 

I        Table  of  numbers,  sqnares,  cubes,  square  roots,  and  cabe 
^H     KWU 479 

^1  APPENDIX  B. 

Remarks  on  the  introduction  of  the  slide  principle  in  tools 
and  machines  employed  in  tbe  production  of  moctiinery, 

by  James  Nosmytb 480 

icwing  tbe  increased  perfection  of  the  workmanship ;  ma- 
I  nnal  dexterity  could  not  bavc  effected  those  productions   .     481 
'egeometrioal  figures;  \iie  line,  pUme,  circle,  cylinder, 
aoA  sphere 482 


xlvi  CONTENTS. 

Alt.  Page 

The  dexterity  of  the  hand  and  eye  of  the  workman    .  483        394 

Mechanical  contriyances  for  holdingy  applying^  and  dvreding 

the  motions  of  a  cutting  tool 484         395 

Accession  of  power  hy  the  slide  rest  principle     ...      —        396 
Comparison  of  this  power  to  that  of  the  steam  engme  itself .     486        397 
Hhistration  of  the  figures  in  the  turning  lathe      .         .         .     487         398 
niustration  of  the  slide  rest  principle. — Fig.  1  representing 
the  system  of  hand  turning  hefore  the  introduction  of  the 

rest 488         398 

Tool  holted  firmly  to  the  rest,  which  slides  along  at  the  com- 
mand of  the  machinist,  illustrated  hy  Fig.  2.  .  .     489         399 
Method  of  communicating  motion  by  the  hand  of  the  work- 
man, or  by  the  introduction  of  the  self-acting  principle, 

explained  in  Fig.  3 489        399 

Application  of  this  operation  where  neither  the  hand  nor  the 

eye  of  the  workman  can  be  used 490         401 

The  mechanical  means  of  operating  on  the  most  ddicate  or 
most  ponderous  masses  of  matter  by  means  of  the  slide 
rest,  are  the  results  of  the  late  Henry  Maudsky's  enthu- 
siastic devotion  to  mechanical  science     ....     491         401 
Application  of  the  slide  rest  to  other  important  processes  in 

constructiye  science 492         402 

The  planing  machine  explained :  it  enables  workmen  to  pro- 
duce improved  tools 494-496  403,  404 

Figure  4  represents  the  general  arrangement  of  parts  exist- 
ing in  the  planing  machine 497-501  404-407 

To  the  slide  rest  we  are  indebted  for  the  planing  machine    .     500         406 
Also  in  the  screw-cutting  machine  we  have  simply  a  slide 

rest     This  is  illustrated  by  Fig.  5 501         406 

Again,  in  the  case  of  the  wheel-cutting  machine  we  have  the 

slide  rest  in  full  existence,  as  is  shewn  in  Fig.  6.     .     502-^04  407-410 
Observations  respecting  the  fbnn  of  tools  employed  for  turn- 
ing and  planing  iron,  brass,  &c«,  together  with  remarks  on 
the  hardening  and  tempering  of  such  tools  505         410 

Disgimms  illustrating  the  fbius  of  tools  for  planing,  or 

shaving  metal,  &e. —        411 

F^.  I.  Stfongth,  but  not  acuteness -^        413 

Fig.  8.  Acuteness,  but  not  sUrfo^ —        413 

F^.  3.  Aoutenoss  and  strc^ngth —        413 

Good  therefore  for  pbnittg  and  laming      ....      —        414 
lUustratien  applied  to  the  use  of  the  joiner's  plane  —        415 


CONTENTS.  Xlvii 

An.  Page 

Abo  in  the  fonns  of  drilk 505        416 

Explanation  of  a  tool  gauge,  to  ascertain  whether  any  tool 

be  ground  or  formed  to  the  proper  angle  .  .  .  _  417 
This  gauge  will  answer  for  every  kind  of  planing  or  turning 

tool  whatever 

Oeneral  explanation  of  the  plates 418-469 

Ihdbz    ..,•..         ....  471 


ON 


THE    TEETH    OF    WHEELS, 


ESSAY  I. 


PREFACE. 

Led  from  situation,  as  well  as  curiosity,  to  attend  ver}^  mi- 
nutely to  some  parts  of  practical  mechanics,  one  of  the  ob- 
jects which   early  attracted  the  notice  of  the  author  of 
the  following  short  Essay,   was  the  figure  of  the  Teeth 
of  \^Tieels.     He  observed,  that,  in  forming  these  teeth, 
workmen  followed  rules  for  which  they  could  assign  no  sa- 
tisfectory  reason.     Nor  did  he  then  find  in  books  the  in- 
formation he  wanted :  the  subject  seemed  to  him  to  require 
a  detail  and  simplification,  which  no  English  writer,  with 
whom  he  was  acquainted,  had  given  it.     Afterwards,  in- 
deed, he  found  that  some   French   mathematicians   had 
treated  it  with  much  attention.     But  their  works,  though 
sufficiently  clear  to  those  who  have  studied  mathematics, 
are  too  abstract  to  be  of  general  utility.     In  the  following 
Essay,  therefore,  such  an  elucidation  of  the  subject  has 
been  attempted,  as  might  render  it  plain  to  the  operative 
mechanic — an  object,  which  will  appear  the  more  import- 
ant, the  more  we  consider  the  great  variety  of  useful  pur- 
poses to  which  wheel-work  is  applied. 

d 


1  PREFACE.  QeSSAY    I. 

De  La  Hire  and  Camus  are  the  two  French  writers, 
who  have  treated  most  extensively  this  branch  of  mechan- 
ics.— From  the  work  of  the  latter,  who  has  written  more 
accurately,  and  more  fully,  the  author  has  borrowed 
largely ;  nor  has  he  scrupled  to  take  from  others,  whatever 
he  found  to  suit  his  purpose,  and  to  make  the  fullest  use  of 
the  communications  of  his  friends. 

Of  the  methods  followed,  it  will  be  sufficient  to  remark, 
that  the  subject  naturally  suggested  these  two  general  di- 
visions— First,  the  Principles  of  the  Configuration  of  the 
Teeth  of  Wheels : — Secondly,  the  application  of  these  to 
practice. 

The  first  chapter  contains  the  Principles — the  second, 
their  Application,  with  certain  modifications — 1st,  to 
Spur  GeaVy  under  which  arc  comprehended  the  Wlieel 
and  Trundle ;  the  Wfteel  and  Pinion ;  the  internal 
Pinion^  and  the  Rack  and  Pinion. — And,  2dly,  to  Bevel 
Gear. 

A  third  chapter  is  added,  which  contains  a  manner  of 
forming  Spur  Wheels^  upon  principles  somewhat  diflFerent 
from  those  considered  in  the  preceding  chapter. 

In  the  following  pages,  no  pretensions  are  made,  either 
to  invention  or  profound  investigation.  The  writer  has 
studied  perspicuity  alone,  and  will  have  completely  at- 
tained his  object,  if  he  has  only  been  fortunate  enough  to 
give  such  a  view  of  the  various  kinds  of  teeth,  as  will  en- 
able the  artist  to  form  some  judgment  of  their  respective 
merits,  and  to  execute  any  of  them  with  accuracy  and  ease. 
For  this  purpose  it  has  been  his  aim  to  divest  every  part 
of  the  subject  of  obscurity,  and  to  accommodate  it  to  those 
who  possess  not  the  advantages  of  a  mathematical  educa- 
tion. But  ho  is  far  from  saying,  that  they  will  not  find 
some  difficulties,  particularly  in  the  first  chapter ;  nor  will 
they,  i)erhai>8,  fully  understmid  the  truths  it  contains,  till 
they  SCO  their  relation  to  practice  pointed  out  in  the  sc- 


BSSAT   1.3  PBEFACE.  11 

cond.  He  found,  lihat  without  becoming  exceediagly  pro- 
lix, there  was  no  aToiding  the  use  of  some  mathematical 
terms,  but  of  these  he  has  given  definitions,  either  as 
the  terms  themselyes  occur,  or  at  the  conclusion  of  the 
Essay*. 

*  This  Pre&ce  was  written  seycnJ  years  before  the  translation  of  Camus 
mB  pabliahed. 


1 


ErmrriONsr 


1.  When  two  toothed  whcek  act  upon  one  another,  the 
greater  is  called  the  IV/teel,  and  the  lesser  the  Pinion. 


2.  Instead  of  the  pinion,  the  trundle  is  sometimes  used, 
Bach  as  is  here  represented.     It  is  likewise  known  by  the 
s  of  lantern  and  wallower. 


3  pinions  and  trundles  are  employed  for  the  same 
purposes,  when  the  action  of  two  wheels  is  spoken  of,  in 
general,  the  trundle  is  comprehended  under  the  name  pinion. 


4,  The  teeth  of  wheels  and  of  pinions,  are  comprehended 

r  the  general  term.  Teeth.     Wliere  the  teeth  are  of 

jhe  same  piece  with  the  body  of  the  wheel,  they  are  called, 

roperly,  faef/t ;  when  they  are  each  of  a  particular  piece, 

y  aru  called  cogs.     Tiie  teeth  of  pinions  are  called  leaoes, 

and  those  of  a  trundle  staves. 


2  GENERAL  DEFINITIONS. 

V. 

5.  Whea  the  action  of  wheels  is  spoken  of  in  general, 
under  the  dame  teeth,  are  comprehended  teeth,  (properly  so 
called,)  cogs,  leaves,  and  staves, 

VI. 

6.  The  straight  line  bf,  which  joins  the  centres  bf, 
of  a  pinion  and  wheel,  which  act  together,  is  called  the 
line  of  centres. 


VII. 

7.  When  the  line  of  centres  bp  is  divided  into  two 
parts,  A  B,  A  F,  proportional  to  the  number  of  the  teeth 
in  the  wheel,  and  in  the  pinion,  these  two  parts,  a  b,  a  p, 
are  named  proportional  radii. 

It  may  be  proper  in  this  place  to  show,  in  what  manner 
the  line  of  centres  is  to  be  divided  in  the  proportion  of 
the  number  of  teeth  in  the  wheel  and  pinion  ;  and  for  this 
purpose,  we  shall  denote  the  length  of  the  line  of  centres 
by  I;  the  number  of  teeth  in  the  pinion  by  p,  and  the 
number  of  teeth  in  the  wheel  by  w;  then  by  the  definition, 


GENERAL  DEFINITIONS.  S 

the  line  /  is  to  be  divided  into  two  parts,  having  the  ratio  of 
ptow. 

Let  or  =  the  lesser  segment,  and  y  =  the  greater. 
Then  we  have 

py  =  WX9  and  x  +  y  =  1; 
M  by  diAioi  aad  tr«»p»iti«.,  we  obtaL 

^= ,  and^  =  /-ir/ 

P 
and  by  comparing  these  values  of  ^,  we  get 

(p  +  w)x  =  p  I9  and  therefore  it  is  a:  =— :£ — . 
^  p  +  w 

tin    7 

and  in  like  manner  it  is  y  =  . 

RULE 

For  the  proportional  radius  of  the  pinion.  Multiply  the 
length  of  the  line  of  centres  by  the  number  of  teeth  in  the 
pinion,  and  divide  by  the  number  of  teeth  in  both  the 
wheel  and  pinion. 

For  the  proportional  radius  of  the  wheel.  Multiply  the 
length  of  the  line  of  centres  by  the  number  of  teeth  in  the 
wheel,  and  divide  by  the  number  of  teeth  in  both  the  wheel 
and  pinion. 

vin. 

8.  If  from  the  centres  b  f  are  described,  with  the  pro- 
portional radii,  the  circles  xa,  ra;  these  circles  represent 
two  cylinders,  which  touch  in  the  point  a  as  if  they  had 
teeth  infinitely  small,  or  as  if  one  of  them  were  conducted 
by  the  other  by  contaction  only.  These  circles  I  shall  call 
proportional  circles  ;  or,  as  they  are  termed  by  millwrights, 
pitch  lines. 

IX. 

9.  The  right  lines,  b  k,  f  q,  drawn  from  the  centres  of 
the  pinion  and  wheel,  to  the  extremities  of  their  respective 
teeth,  are  called  real  radii. 

b2 


1  -^ 


CHAPTER  I. 

OF   THE   PRINCIPLES   OF   THE    CONFIGURATION   OF   THE 

TEETH    OF   WHEELS. 

10.  In  the  construction  of  machines,  the  proper  forma- 
tion of  the  teeth  of  wheels  is  an  object  of  much  importance. 
Though  experience  may  often  enable  the  merely  practical 
mechanic  to  approach,  in  this  respect,  to  some  degree  of 
perfection,  yet,  being  ignorant  of  principle,  his  work  is 
always  conducted  with  uncertainty,  and  he  generally  pro- 
duces machines  expensive  in  working,  and  defectiye  in  re- 
gularity, eflFect,  and  duration. 

For  when  the  acting  parts  of  a  machine  are  not  truly 
formed,  it  may  be  so  loaded  as  just  to  be  in  equilibrio  with 
its  work  in  the  most  favourable  situation  of  its  parts,  but 
when  it  changes  into  a  less  favourable  situation,  the  machine 
will  stop,  or  at  least,  stagger,  hobble,  or  work  unequally. 

The  best  figure,  therefore,  which  can  be  given  to  the 
teeth,  is  that  which  shaU  cause  them  always  to  act  equally 
and  similarly,  in  situations  equally  favourable,  and  which 
shaU  consequentiy  give  the  machine  the  property  of  being 
moved  uniformly  by  a  power  constant  and  equal ;  or,  in 
other  words,  ensure  an  uniformity  of  pressure  and  velocity. 

Were  the  teeth  of  wheels  infinitely  small,  their  action 
would  be  regarded  as  that  of  cylinders,  simply  touching, 
hanng  the  property  required.  The  finite  and  sensible 
teeth  gi^-en  to  wheels  will,  therefore,  be  of  the  most  advan- 
tageous figure,  when  one  wheel  conducts  another,  as  if  they 
simply  touched ;  or  when  their  pitch  lines  have  in  every 
part  of  their  revolution  etjual  velocities. 


I 

I 


I 

I 


[  CHAP.  I.]  ON    THE    TEETH    OF    WHEELS.  5 

That  teeth  have  this  property,  when  formed  in  a  certain 
manner,  will  be  evident  from  the  foUowinff  proposition  and 
its  connections'. 

PKOPOSITION. 

1 1.  When  teeth  are  of  such  a  form,  that  a  perpendicular 
H  E 1 1  (Fig.  2.  p.  2.)  drawn  to  the  tangent  to  the  edge  of  the 
tooth  in  tho  point  of  contact  e,  cuts  the  line  of  centres  at 
the  termination  a,  of  their  proportional  radii,  their  pitcft 
lines  shall  have  in  corresponding  places,  equal  velocities, 
whether  the  wheel  drives  the  pinion,  or  the  pinion  the 
wheel ;  that  is  to  say,  that  they  will  move  each  other  as  if 
they  merely  touched  t. 

The  manner  of  fonning  teeth  of  wlieels  here  refciTcil  to  by  oar  Author, 
would  ensure  an  equable  communicatiou  of  power  or  motion  in  the  imogin- 
when  the  rubbiug  parts  have  no  seusibJe  friction ;  but  in  no  other; 
except  it  he  possible  to  contrive  a  prat'ticable  form  for  teeth  baring  the  pro- 
perty of  Teoderiiig  the  friction  uniform  during  the  action  of  each  pair  of 
loelJi :  this  has  not  yet  been  accomplished.  Hence  it  appears  that  practical 
men  have  not  without  reason  been  doubtful  of  tho  adrantages  of  the  kind 
of  t«eUi  proposed  by  mathematical  i^Titcrs ;  for  that  the  iriction  of  teeth  is 
sot  uoifonD,  Dr.  Yonng  has  proved  in  a  letter,  which  forms  a  Tolnable  port 
of  thisEeaay,  (see  Art.  66 — 71.)  And  we  have  o  practical  proof  of  the 
>uie  thing  b  the  unequal  wear  of  teeth,  (sec  Art.  40.)  The  best  means 
of  K<raidiiig  the  inequality  produced  by  friction  seems  to  be,  to  make  the 
h>Mh  "oi  smalt  and  as  numerous  as  is  consistent  with  strength  and  dura- 
bfli*T,"  (Art.  65.)  These  limits,  with  reqiect  to  strength  and  durahiiity,  I 
will  endeBTour  to  establish  In  the  additions  to  Art.  121,  and  those  following 
it.  And  since,  in  adopting  the  principle  of  short  teeth,  tlie  curved  surface 
of  cad  tootti  will  become  so  small  tljat  a  circular  arc  may  be  employed  in- 
stead of  the  proper  curve,  we  slmll,  in  the  additions,  point  out  the  mode  of 
dMcribing  ciicnlar  arcs  to  answer  this  purpose. 

+  For  tint  manner  of  drawing  this  perpendicular,  see  Art.  18. 

{  This  bnug  a  Fundamental  proportion,  it  is  of  importance  that  it  should 
'ie  •rcQ  understood;  wesbnll  therefore,  in  this  note,  attempt  a  popular  illus- 

ition  of  it. 

It  u  deoioiislroble  tliat  iho  line  H  B  (Pig.  2.)  has  the  same  proportion  to 
the  line  I  r  which  a  b  Ims  to  a  p.  For  tunce  b  b  and  f  i  are  paroUcI,  each  of 
them  being  perpendicular  to  u  i ;  it  follows  that  the  triangles  asm  and  a  f  i 


6  ON  THE  TEETH   OF   WHEELS.  []CHAP.  1. 

We  shall  now  proceed  to  show,  that  the  epicycloid  gives 
the  property  to  the  teeth  of  wheels  required  in  the  preced- 
ing proposition,  and  shall  begin  with  some  definitions  re- 
specting that  curve. 

Before  we  proceed  with  our  Author,  it  will  be  an  advan- 
tage to  examine  this  proposition  more  particularly. 

12.  Let  AH  (Fig.  2.)  be  the  direction  of  the  force  of  the 
wheel  to  turn  the  pinion,  and  b  h  a  line  perpendicular  to  a  h, 
drawn  to  the  centre  of  motion  b.  The  eflFect  of  the  force 
to  turn  the  pinion  will  be  directly  as  the  distance  of  its 
direction  from  the  centre  of  motion,  or  as  hb.  Also,  the 
angular  velocity  generated  will  be  inversely  as  the  distance 

of  the  direction  from  the  same  centre,  or  as  — .      *  Conse- 

HB 

« 

quently,  the  quantity  of  motion  communicated  to  the  pinion 
is  as  — ;  that  is,  in  an  invariable  ratio ;  but  by  the  same 

HB 

reasoning  it  may  be  proved  that  the  force  of  the  wheel  at  a 
is  invariable ;  and  therefore,  the  pinion  will  be  moved  in 
the  same  manner  as  if  it  were  moved  by  contact  at  a,  when 
HA  is  perpendicular  to  the  common  tangent  of  the  surfaces 
in  contact  at  e. 

The  same  may  be  proved  when  the  pinion  drives  the 
wheel.  But  this,  as  well  as  the  more  detailed  investiga- 
tions  of  Camus  (on  the  Teeth  of  Wheels,  Art.  521.)  and 
his  followers,  neglects  the  eflfect  of  friction.  Let  the  eflFect 
of  the  friction  of  the  surfaces  be  represented  by  :r,  when 
the  pressure  and  velocity  of  these  surfaces  are  each  equal 

are  similar,  or  equiangular ;  but  the  sides  about  the  equal  angles  of  equian- 
gular triangles  are  proportional :  therefore,  it  is 

HB  :  IF  ::  ab  :  af. 
Now  let  us  suppose  h  b  and  i  f  to  be  levers,  and  h  i  a  string,  the  one  lever 
pressing  from  the  other,  would  act  upon  it  with  just  the  same  force  that  the 
pinion  and  wheel  do  at  the  point  a,  where  the  pitch  lines  touch ;  or,  in 
other  words,  as  if  the  circle  x  acted  on  the  circle  r,  by  means  of  a  string,  as 
pulleys  do  on  each  other  by  a  band. 


I 


CH-IP.  1.3  ON    THE   TEETH    OF    WHEELS, 

to  Hnily,  or  1 ;  then  the  ratio  will  be 

1 


H  B  (  i   —  J") 


;  which  is  invariable  only  when  the  friction  is  invari- 
able. But  when  the  teeth  are  very  short,  and  formed  so 
that  the  motion  would  be  uniform  were  the  friction  uniform, 
it  is  perhaps  the  best  practical  method  of  forming  teeth. 

DEFINITIONS. 


13.  If  upon  the  same  immoveable  plane  are  placed  two 
circles,  CNP,  calmk,  (Fig.  3  and  4.)  which  touch  each 
other  in  the  point  c,  and  the  former,  with  a  supposed  style 
or  tracer  in  its  circumference  at  the  point  c,  is  made  to  re- 
volve round  the  circumference  of  the  latter,  the  style  c, 
during  the  revolution,  will  describe  upon  the  plane  calmk 
Ibe  curve  cegdk,  which  is  called  an  epicycloid.  The 
fpic^cloid  thereibre,  is  a  curve  generated  by  a  point  in  one 
circle  revolving  about  another,  either  on  the  concavity  or 
wnTCxity  of  its  circumference,  and  thus  it  differs  from  the 
I  cycloid,  which  is  generated  by  the  revolution  of  a 

s  along  a  straight  line.  The  cycloid,  however,  has 
me  times  been  assimilated  with  the  epicycloid,  by  con- 
lering  the  straight  line  as  the  circumference  of  a  circle  of 
I  the  tUamcter  is  infinite. 


\  1-k  The  circle  c  n  p,  which,  in  revolvmg  describes  the 
picycloid,  is  called  the  generating  circle  of  the  epicycloid, 

1  the  arc  calmk  of  the  immoveable  circle,  upon  which 

!  generating  circle  revolves,  is  called  the  hase  of  the 

Ofcloid. 
\  Epicycloids  are  distinguished  into  two  sorts,  exterior 

I  interior. 


8 


ON   THE  TEETH   OF  WHEELS.  [CHAP.  L 


Fio.  3. 


III. 


15.  When  the  generating  circle  revolves  without  the 
circle  of  its  hase,  as  in  Fig.  3,  the  epicycloid  is  called  an 
exterior  epicycloid. 


Fio.  4. 


And  when  the  generating  circle  roUs  within  the  circle  of 
its  hase,  as  in  Fig.  4,  the  epicycloid  is  called  an  interior 
epicycloid. 

COROLLARIES. 

I. 

16.  As  the  generating  circle  in  revolving  from  its  first 
situation,  c  n  p,  to  different  portions,  ae f,  l  gh,  &c.  ;  applies 


OS    THE   TEETH   ( 


9 


lluccesfflTely  all  the  parts  of  its  circumference  to  those  of  its 

is  evident  the  hsse,  c  a  l  m  K,  of  the  epicycloid  h 

I  the  circumference  of  the  generating  circle  c  n  p  c, 

and  each  such  portion,  as  c  a,  or  c  l,  &c.,  of  the  base,  is 

equal  to  each  part  ea,  or  gl,  of  the  circumference  of  the 

I  generating  circle. 

Hence  a  method  of  drawing  the  epicycloid,  by  deacrih- 

ing  the  circles  aef,  lgh,  &c.,  which  have  all  the  same 

ndii  as  the  generating  circle  cnp,  and  touch  the  base 

I  CALHK  in  anypoints  a,  l,  &:c.;  and  by  making  the  length 

I  of  the  arcs  ae,  lg,  &c.,  taken  from  the  points  of  contact 

I  with  the  base,  equal  to  the  arcs  ac,  cl,  &c*. 

Haring  thus  determined  as  many  points,  E,  G,  &c.,  as 

MY  be  necessary,    the  curve  c  e  c  d  k,    which  shall  pass 

ftbrough  them  and  the  point  c,  where  the  supposed  style 

i  generating  circle  was  supposed  to  begin  its  tract, 

I  shall  be  an  epicycloid  t. 

II. 

I"-  \Vhen  the  generating  circle  cnp  revolves  within  the 
circle  of  its  base,  (Fig.  5,)  and  has  for  its  diameter  the 
radius  Bc  of  its  base,  the  point  c,  the  place  of  the  style 
wing  the  revolution  of  the  generating  circle,  will  always 
mtiQue  in  the  diameter  c  B  k.  Hence  the  epicycloid  de- 
irihed  6y  the  style  c  is  a  straight  line,  and  a  diameter  of 

*  Id  pncticc,  tlik  is  most  cosily  done,  and  witli  suifident  accuracy,  by 
g  cneli  arc  of  tbe  base,  as  at  A  c,  into  a  number  of  Bmall  equal  parts, 
d  bv  Betting  off  tbe  same  number  upon  each  arc  of  tie  generating  circle. 


1 1  To  diKW  tlie  epicycloid  mecbanically,  make  the  circle  of  the  boee  and 
M  genemtiDg  circle  of  wood,  and  linving  lixod  a  tracer  in  the  circumference 

Ctke  gmeniting  circle,  let  the  base  remain  at  rest,  and  the  tracer,  during 

a  rolHag  nf  the  genemting  circle,  will  draw  an  epicycloid.     In  order  to 

o  ciidcs  move  with  more  accuracy,  a  small  piece  of  tape  may  have  one 

F  its  ends  noded  to  the  circumference  of  the  one  circle,  and  the  olher  cud 

B  the  oiber  drde. 


10  ON   THE   TEETH   OF   WHEELS.  [CHAP.  I. 

the  circle  of  its  hose*;  and  the  circumference  of  the  gene- 
rating circle  c  n  p  being  half  that  of  the  base,  the  commence- 
ment c  and  termination  k  of  the  epicycloid,  must  divide 
the  circumference  of  the  base  into  two  equal  parts,  and  the 
diameter  ab  of  the  generating  circle  being  half  that  of  kc 
of  the  base,  when  the  generating  circle  is  in  the  middle  of 
its  progress,  the  point  c  must  be  in  the  centre  of  the  circle 
of  the  base  c  (or  coincident  with  b)  ;  hence  we  have  a  point  c 
at  the  origin  b,  in  the  middle,  and  k  at  the  end  of  the 
epicycloid,  which  all  lie  in  c  k,  the  diameter  of  the  base, 
and  the  whole  epicycloid  may  be  considered  as  coinciding 
with  CK,  the  diameter  of  the  baset. 

Fig.  5. 


III. 

18.  When  the  generating  circle  of  the  epicycloid,  as  in 
Fig.  6,  is  in  any  position,  a,  e,  b,  touching  the  circumfer- 
ence of  its  base  in  any  point  a,  a  straight  liney  drawn  from 

*  Upon  this  principle  a  parallel  motion  has  been  constructed.  It  is  used 
by  Messrs.  Fenton,  Murray,  and  Wood,  in  some  of  their  smaller  steam 
engines. 

For  a  short  account  of  it,  see  Gregory's  Mechanics,  vol.  ii.  p.  265. 

t  It  would  carry  us  too  far  into  mathematics  for  many  readers,  were  we 
strictly  to  demonstrate,  that  eyery  point  of  the  epicycloid  must  lie  in  the 
diameter  of  the  base ;  what  is  said,  howeyer,  will  satisfy  them  of  the  truth. 
The  mathematical  reader  will  find  a  demonstration  of  this  in  ^^  Cours  de 
Mathematique,  par  Camus,"  iy.  No.  538 ;  or  En^ish  Translation  of  that 
part  which  treats  of  the  Teeth  of  Wheels,  and  from  which  Buchanan  bor- 
rowed laigcly  in  this  part  of  his  work. 

f 


CHAP.   I.^  ON   THE   TEETH   OF   WHEELS.  11 

the  point  of  contact  a  to  the  point  e,  actiuiUy  describing 
Ae  epicycloid^  will  be  perpendicular  to  it. 

This  will  be  evident  by  supposing  the  circles  to  be  poly- 
gons, having  a  great  number  of  sides.  For  when  turning 
on  any  of  the  summits,  the  tracer  describes  a  small  part  of 
a  circle  from  that  summit  as  a  centre,  and  will  consequently 
be  perpendicular  to  it. 

Pio,  G. 


IV, 

19.  Let  us  imagine  in  the  same  plane  three  circles,  r,  x, 
T,  Fig.  7>  which  touch  in  the  same  point  a,  and  which  con- 
sequently have  their  centres,  f,  b,  g,  in  a  straight  line,  and 
are  moveable  round  their  centres  only. 

Suppose  a  style  fixed  in  the  circumference  of  the  circle 
T,  and  that  the  three  circles  are  made  to  turn  by  the  move- 
ment of  one  of  them :  if  we  make  each  of  the  arcs,  ah,  a  c, 
equal  to  a  e,  then  the  style  placed  in  e  shall  have  described 
on  the  plane  of  the  circle  r,  a  portion  cy,  of  an  exterio r 
epicycloid,  and  on  the  plane  of  the  circle  x,  a  portion  h  e 
of  an  interior  epicycloid. 


12 


ON   THE  TEETH   OF   WHEELS*  [^CHAP.  I. 


Pig.  7. 


The  two  epicycloids  J  c  e,  he,  traced  in  the  same  time  hy 
the  style  e,  touch  in  the  point  e.  For  the  straight  line  ae, 
drawn  from  the  point  a,  where  the  generating  circle  y 
touches  its  base  rc,  shall  be  perpendicular  to  the  two  epi- 
cycloids, and  the  straight  line  h  e  shall  touch  the  epicycloid 
in  the  point  e*. 

V. 

20.  Let  us  next  suppose,  that  the  generating  circle  y 
has  for  a  diameter  the  radius  ab  of  the  circle  x,  within 
which  it  is  placed,  and  that  the  three  circles,  r,  x,  y,  touch 
continually  in  the  point  a,  as  in  Fig.  8. 

The  interior  epicycloid  h  e,  which  touches  the  exterior 
c  E,  shaU  be  a  straight  line  directed  towards  the  centre  b 
of  the  circle  x.  Art.  16,  and  consequently  a  portion  of  the 
radius  b  h,  which  shall  always  touch  the  exterior  epicycloid 
c  E  in  the  point  e,  where  it  shall  be  met  by  the  perpendicu- 
lar AE. 


*  Because  any  triangle  wliich  can  be  inscribed  in  a  semicircle,  is  a  right- 
angled  triangle.  For  manner  of  drawing  a  perpendicular  on  the  end  of  a 
line,  see  supplementary  definitions,  Art.  83. 


CHAP.  I.]  ON   THE   TEETH    OF   WHEELS. 


13 


Pig.  8. 


Hence  it  follows,  that  when  the  two  circles^  r,  x,  touch 
continually,  and  the  one  causes  the  other  to  turn  by  con* 
tact  at  the  point  a,  if  we  imagine  a  radius  b  h  in  the  circle 
X ;  and  haying  made  a  c  equal  to  a  h,  there  will  be  described 
by  the  point  c,  an  exterior  epicycloid  ce,  which  has  for  a 
generating  circle  y,  the  diameter  of  which  is  equal  to  the 
radius  bh,  this  radius  bh,  during  the  movement  of  the 
circles  r,  x,  shall  always  touch  the  epicycloid  in  the  point  £, 
where  this  epicycloid  shall  be  cut  by  the  straight  line  a  e 
perpendicular  to  its  curve. 

Thus  instead  of  supposing,  that  one  of  the  two  circles 
R,  X,  turns  forward  the  other  by  the  point  of  contact  a,  let 
it  be  supposed,  that  the  one  is  made  to  push  forward  the 
radius  bh,  of  the  circle  x,  by  an  epicycloid  ce  attached  to 
the  circle  a,  and  described  by  the  movement  of  the  circle 
T,  the  diameter  of  which  is  equal  4o  the  radius  b  h. 

One  may  be  able  thus  reciprocally  to  make  the  epicycloid 
c  £,  attached  to  the  circle  r,  push  forward  by  a  radius  b  h 
a  circle  x  ;  and  by  means  of  the  epicycloid  c  e,  and  of  the 


14 


'  THE   TEETH   OF  WHEBLSi 


[chap.  I. 


radius,  bh,  the  two  circles,  r,  x,  may  be  able  to  conduct 
themselves  as  if  put  forward  by  the  point  of  contact  a  *. 

For  suppose  the  radius,  bh,  and  the  epicycloid,  ce,  to  be 
teeth  oi  wheels,  x  and  t  ;  and  the  perpendicular  ae,  from 
the  touching  sur&ces  in  all  situations,  cuts  the  line  of 
centres  at  the  termination  a  of  their  proportional  radii. 
But  we  saw.  Art.  H,  that  when  this  was  the  case,  the  pro- 
portional circles  must  have  equal  velocities. 

It  is  principaUjf  from  this,  that  we  shall  deduce  the  best 
Jigure  which  can  be  given  to  the  teeth  of  wheels  and  pinions, 
when  one  part  of  the  wheel  and  pinion,  or  of  both,  ought  to 
be  a  straight  line  tending  to  the  centre  of  such  wheel  or 
pinion. 

VI. 

@1.  If  in  the  same  plane  we  have  hut  two  circles,  r,  t. 
Fig.  10,  which  touch  in  the  point  a,  and  if  the  movement 
of  the  one  communicate  itself  to  the  other,  hy  this  point  of 
contact,  any  point  e  of  the  circumference  of  the  circle  t, 
describe  upon  the  plane  of  the  moveable  circle  r,  an  epicy- 
cloid CE. 


Fio. 

*  To  be  Batiefied  of  this  experimentally, 
make  aay  two  drcles  of  wood,  aa  in  Fig.  fi ; 
to  the  ciTcomference  of  one  of  them  a,  fix  * 
piece  of  wood  b,  fonned  into  an  epicycloid, 
generated  by  a  circle  half  the  diameter  of  c 
upon  A  as  s  base. 

On  the  circle  c,  draw  the  line  c  d,  and  cnt 
out  the  part  bounded  by  that  line  and  c  b. 

If  you  cause  one  of  the  circles  to  move 
tke  other  by  the  parts  b,  c  d,  both  circles  will 
bsTO  the  same  velocity;  as  may  be  aacer- 
l^ed  by  putting  a  mark  oppodte-any  pomt 
in  the  circumference  of  each  circle  before  they  be^n  to  move,  and  anotber 
after  they  stop,  and  the  distance  between  ^rtiich,  measuring  by  the  arcs,  wiQ 
be  found  equal. 


CHAF.  lO 


OK  THE  TEETH  OP  WHEELS. 

Fio.  10. 


Suppose  this  epicycloid  attached  to  the  circle  r,  it  (the 
epicycloid)  shall  conduct  the  circle  y,  pushiog  it  round  by 
the  point  e  of  its  circumference,  in  the  same  manner  as  the 
circle  r  might  conduct  the  same  circle  t  in  commimicating 
motion  to  it  by  the  point  of  contact  a. 

And  in  like  manner,  the  point  E  of  the  circumference  of 
the  circle  y,  turns  the  circle  r,  in  pushing  it  by  the  epicy- 
cloid CE,  supposed  to  be  attached  to  b,  in  the  same  way 
that  the  circle  y  would  conduct  the  circle  r  in  communicat- 
ing its  motion  by  the  point  of  contact  a  *. 


*  The  experiment  to  prove  this  is  umilar 
to  the  fonner,  bnt  with  this  diSerencc,  that 
in  the  circuinfereiice  of  one  of  them,  a,  is 
fixed  a  fine  needle,  which  is  made  to  act 
against  a  piece  of  wood,  fonned  into  an 
epicycloid,  fixed  upon  the  other,  b,  which 
epicycloid  is  generated  by  a  upon  b  as  a 
1»se. 


16  ON   THE   TEETH   OF   WHEELS.  [CHAP.   I. 

The  same  mode  of  proof  applies  here  that  did  to  the 
corollary  immediately  preceding. 

This  last  corollary  enables  us  to  determine  the  best 
figure  which  can  be  given  to  the  teeth  of  wheels j  when  the 
pinion  shaU  be  a  trundle  composed  of  staves. 

We  shall  likewise  determine  from  it  the  most  advan- 
tageous figure  which  can  be  given  to  the  teeth  of  a  pinion^ 
when  the  wheel  shall  have  staves  in  place  of  teeth. 

22.  In  addition  to  the  properties  of  the  epicycloid  men- 
tioned above,  there  are  several  others  of  a  curious  and  sci- 
entific nature,  which  may  perhaps  be  not  improperly  intro- 
duced  in  this  place,  although  they  may  not  be  immediately 
applicable  to  the  construction  of  the  teeth  of  wheel  work. 

1.  If  the  generating  and  quiescent  circle  have  to  each 
other  any  commensurable  ratio,  then  is  the  epicy- 
cloid thus  generated  both  rectifiable  and  quadrable ; 
that  is,  both  its  length  and  area  are  exactly  deter- 
mmable. 

2.  If  the  generating  and  quiescent  circles  are  incom- 
mensurable with  each  other,  then  the  epicycloid  is 
unquadrable,  but  it  is  still  rectifiable ;  that  is,  the 
area  in  this  case  cannot  be  foimd  in  finite  terms, 
although  the  length  of  the  curve  is  exactly  assign- 
able. 

23.  To  these  we  may  also  add  the  following  rules  for 
finding  the  lengths  of  epicycloidal  curves,  and  the  areas 
which  they  enclose. 

RULE  L 

As  the  semidiameter  of  the  quiescent  circle,  is  to  the 
sum  of  the  diameters  of  the  two  circles,  so  is  double  the 
versed  sine  of  the  arc  of  the  generant,  which  has  passed 
over  any  portion  of  the  quiescent  circle,  to  the  length  of  the 
epicycloidal  arc  generated  by  the  point  which  touched  the 
quiescent  circle  or  base  at  the  beginning  of  the  motion. 


CHAP.   I.]  ON   THE   TEETH    OF   WHEELS.  17 

When  the  whole  arc  is  required,  the  versed  sine  becomes  the 
diameter  of  the  generant  The  length  of  any  arc  of  an  in- 
terior  epicycloid  is  found  in  a  similar  manner,  only  using 
the  difference  of  the  diameters  in  the  second  term  of  the 
proportion  instead  of  the  sum. 

RULE  II. 

To  find  the  area  of  an  epicycloid ;  it  is,  as  the  radius  of 
the  quiescent  circle  is  to  three  times  that  radius,  plus 
twice  the  radius  of  the  generant,  so  is  the  circular  segment 
AE,  to  the  epicycloidal  sector  aec.  Or,  so  is  the  whole 
area  of  the  generant,  to  the  whole  area  of  the  epicycloid. 
This  rule  applies  to  both  the  exterior  and  interior  epicy- 
cloid. 

A  general  proposition  for  the  area  of  all  cycloids  and 
epicycloids  is  given  by  Dr.  Halley,  and  is  as  follows,  viz. : 
That  the  area  of  a  cycloid  or  epicycloid,  either  primary, 
curtate  or  prolate,  is  to  the  area  of  its  generating  circle,  as 
the  ram  of  double  the  velocity  of  the  centre,  and  velocity 
of  the  circular  motion,  to  the  velocity  of  the  circular  mo- 
tion. 

The  same  proportion  holds  good  m  reference  to  any  parts 
generated  in  those  curves,  and  the  analogous  segments  of 
tte  generating  circle. 


CHAPTER  IL 

OF   THE   APPLICATION   OF   THE   PRINCIPLES   OF   THE 
CONFIGURATION   OF  THE   TEETH   OF   WHEELS. 

24.  Having  endeavoured  to  show,  tha4;  an  epicycloid  is 
a  curve,  whereby  two  circles  may  conduct  themselves  as  if 
put  forward  by  the  simple  contact  of  their  circumferences, 
I  shall  now  attempt  a  practical  explanation  of  this  curve, 
in  giving  the  best  form  to  the  teeth  of  wheels. 

SECTION  I. 

OF  8PUB  GSABS. 

25.  By  Spur  Oeers  is  understood  wheels  acting  toge- 
ther, and  in  the  same  plane,  with  their  axes  parallel ;  under 
this  head  the  wheel  and  trundle  come  first  to  be  con- 
sidered. 

OF  THB   WHBBL  AKD  TBUNDLB. 

26.  To  determine  the  figure  of  the  teeth  of  the  wheel, 
which  depends  always  upon  that  of  the  staves  of  the  trun- 
dle, we  shall  first  suppose  the  staves  to  be  indefinitely  small, 
and  represented  (Fig.  12)  on  the  end  of  the  trundle  by  the 
points.  A,  E,  H,  &c. :  when  we  have  found  the  figure  of  the 
teeth  proper  to  conduct  the  indefinitely  small  staves,  (which 
are  used  for  demonstration  only,)  we  shall,  by  means  of 
that  figure,  trace  the  true  form  which  should  be  given  to 
the  teeth  of  wheels  to  conduct  trundles  with  cvlindric  staves 
of  some  magnitude.  Thus  the  solution  of  this  case»  na- 
turally di\-ides  itsi^f  into  twi>  parts. 


CBAP.   11.3  OS   THE   TEETH    OF    WHEELS.  19 

Fm.  12. 


TO  Tina   THB  nOURB  OF   THB  TEETH    WBBK   THE  STAVES  ARE 
IHDBFINITBLY   SHALL. 

37.  Draw  the  proportional  circles,  c  ac  and  e  ak,  and 
divide  each  of  them  into  the  number  of  equal  parts  which 
it  should  have  of  teeth'. 


*  Thii  opnatioa  ia  called  by  millwrights  tMinff  tff  the  pitch.  By  tbe 
|Htcli  is  undentood  the  dlstBoce  between  the  centres  of  two  condgaoiu 
teeth. 


so  ON   THE   TEETH   OF  WHEELS*  [CHAP.   II. 

We  have  seen*,  if  the  circle  cac,  which  touches  the 
circle  e  a  e,  have  attached  to  its  circumference  an  epicycloid, 
c  E,  described  by  the  point  e  of  the  circumference  of  the 
circle  e  a  e  rolling  upon  the  circle,  c  a  c,  the  epicycloid  con- 
ducts the  circle  e  a  e  by  the  point  e,  as  if  conducted  by 
contact  at  a,  and  consequently  the  circumferences  of  the 
two  circles  shall  have  the  same  velocity. 

The  epicycloid,  c  e,  is  then  the  best  figure  which  can  be 
given  to  the  teeth  of  a  wheel  to  conduct  a  trundle,  the 
staves  of  which  are  indefinitely  small,  and  therefore  must 
move  the  stave  e,  in  the  direction  from  a  towards  e,  until 
a  second  stave  arrive,  and  be  taken  in  the  line  of  centres 
by  a  second  epicycloid,  a  b,  which  shall  in  like  manner  con- 
duct this  stave,  a,  until  the  arrival  of  another  stave,  e,  in 
the  said  line:  and  thus  the  other  staves  of  the  trundle 
shall  be  conducted  by  the  other  epicycloids  of  the  wheel. 

Here  it  may  be  observed,  though  perhaps  already  evi- 
dent, that  it  is  the  convex  side  of  the  epicycloid  which 
must  be  used :  for  though  it  be  useful  in  some  machines, 
to  make  the  concave  side  of  a  single  epicycloid  conduct  a 
point  of  a  single  piece  moveable  on  a  centre,  yet  were  a 
number  of  teeth  so  formed,  it  would  be  impossible  for  them 
to  act  on  a  number  of  staves,  for  they  would  be  so  hooked 
and  entangled  as  not  to  move  forward  in  the  smallest  de- 
gree. 

Were  it  wished  that  the  wheel  should  move  the  trundle 
in  both  directions,  it  is  obvious  that  each  tooth  of  the  wheel 
should  have  its  opposite  sides,  c  e,  l  m,  formed  into  equal 
epicycloids. 

As  we  have  supposed  the  staves  of  the  trundle  indefi- 
nitely small,  were  the  teeth  of  the  wheel  also  perfect 
figures,  and  djually  distanced,  there  would  be  no  need  of 
other  than  indefinitely  small  spaces  between  the  adjacent 

""  See  Chip.  L  Article  21. 


[  CHAP.  11.3  ON    THE   TEETH    OF    WHEELS. 


n 


teeth  of  the  wheel ;  but  as  perfect  precision  is  not  to  be 
expected,  a  space  more  or  leas,  such  as  al,  must  be  left 
between  them,  to  enable  the  wheel,  notwithstanding  the  in- 
equalities of  the  teeth  and  staves,  to  move  the  pinion. 

We  have  hitherto  supposed  (he  teeth  uf  the  wheel  con- 
ducted by  the  staves  of  the  trundle,  hut  it  is  evident,  had 
the  teeth  of  the  wheel  the  same  figure,  when  conducted  by 
the  staves,  the  wheel  and  trmidle  would  retain  thp  property 
of  moving  with  the  same  velocity.  It  may  only  be  observed, 
that  the  staves  of  the  tnmdle  conduct  the  teeth  of  the 
wheel  in  approaching  the  line  of  centres,  while  the  teeth 
of  the  wheel  conduct  the  staves  of  the  trundle  in  their  pro- 
gress from  that  line*. 


[XO  nUD    THE    PIOURBS  OV   1 
OP  TBB  TBUtlBLB  A 


E   TEETH   OP  THE  WHHKL,  WHEN 
f  CVt-lNDEHS  OP  A   PIN 


I 


28.  Consider  the  trundle  at  first  as  having  infinitely 

IsmaD  staves,  represented  by  the  centres  of  the  staves, 
A,  E,  u,  &c.,  and  trace,  as  above  mentioned,  the  teeth  c  L  p, 
A  Q.s,  &c.  of  the  wlieel,  as  if  it  had  to  conduct  a  trundle 
with  infinitely  small  staves :  observing  to  leave  a  small 
space,  such  as  a  l,  between  all  the  teeth,  in  order  that  they 
may  act  freely. 

Describe,  with  the  radius  of  the  staves,  upon  the  plane 
of  each  tooth,  as  many  small  arcs  as  may  be  convenient, 
ba\'iag  all  their  centres  in  the  two  epicycloids  which  form 
the  teeth. 

Trace,  by  means  of  these  little  arcs,  two  curves,  such  as 
»o,  so,  parallel  to  the  epicycloids,  and  then  you  will  have 
I  inclosed  the  space,  kos,  which  is  the  figure  all  the  teeth  of 
I  the  wheel  ought  to  have  beyond  its  proportional  circle. 

•  Soe  Article  33  of  this  chupter. 


1 


t  THE  TEETH  OF  WHEELS.  (JCHAP.  II. 

Fia,  13. 


For  if  wc  suppose,  that  the  centre  e,  of  a  stave,  is  con- 
ductetl  by  the  tooth  cpl  ;  the  curve  ro,  which  is  parallel 
to  the  epicycloid  ci",  and  which  is  placed  at  the  distance  of 
the  radius  of  the  stave  e.  shall  always  touch  the  circumfer- 
ence of  that  stave. 

Thus  the  cur%-e  ro  shall  conduct  the  eylindric  stave,  as 
if  the  tooth  c  r  i.  cundncted  the  centre  of  that  stave,  and 
conisequently  the  twtlh  ros,  shall  he  a  proper  figure  to 
wnduct  the  trundle,  with  eylindric  staves. 

The  eunvil  ^larts  of  the  teeth  of  the  wheel,  being  deter- 
mined as  above,  the  spac«s  ts  >&>  &c.  should  be  cut  out. 


[  CHAP.    II.]  ON    THE    TEETH    OF    WHEELS,  'iS 

in  order  to  admit  that  part  of  the  staves  which  extends  be- 
yond the  proportional  circle  of  the  trundle. 


ma^i 


TO  OESCRIBB   THE   TEETH   C 


'  A    WIIEEI..    FOR  A 
C1BCUL*B   ARCS. 


h 

^H  29.  L«t  c  D  he  the  line  of  centres  ;  e  e  the  pitch  line  of 
^^  the  trundle ;  and  f  f  that  of  the  wheel ;  and  suppose  the 
cpQtre  of  the  stave  a  to  be  in  the  line  of  centres  c  D  ;  then 
place  one  foot  of  the  compasses  in  the  centre  of  the  stave 
A,  and  describe  the  arc  be,  which  is  the  form  of  the  tooth. 
The  part  of  the  teeth  of  the  wheel,  within  the  pitch  line, 
I  may  be  described  with  circular  arcs  as  in  the  figure. 

Teeth  formed  in  this  manner  will  not  sensibly  differ  from 

Ithose  described  according  to  the  principles  laid  down  in  the 

ding  articles,  when  the  length  of  each  tooth  is  not 

f  greater  than  is  necessarj-.    The  reader  will  easily  perceive, 

that  the  radius  for  describing  the  teeth,  is  equal  to  the 

pitch  diminished  by  half  the  diameter  of  the  stave ;  and 

I  aUo  that  the  centres  of  those  arcs  will  always  be  in  the 

I  proportional  circle,  or  pitch  line  of  the  wheel. 

^me  authors  have  imagined  that  the  friction  of  the 
I  wheel  and  trundle  might  be  reduced  Ity  making  the  staves 


Q4t  ON   THE   TEETH   OF   WHEELS.  [CHAP.  II. 

revolve ;  but  it  could  not  be  effected  so  far  as  to  balance 
the  extra  labour  of  construction,  and  where  the  strain 
would  be  considerable,  it  would  become  quite  impracticable. 
(See  Emerson's  Mechanics,  prop.  119,  rule  9 ;  and  Trans- 
actions of  the  Society  of  Arts,  voL  xxxv.  p.  128.) 

Smeaton  appears  to  have  been  very  partial  to  the  wheel 
and  trundle,  when  the  trundle  was  executed  with  cast  iron 
staves ;  these  he  recommended  to  be  of  an  oval  figure,  and 
made  smooth  by  grinding  them.  (Smeaton's  Reports,  voL 
i.  p.  316 ;  voL  ii.  p.  391  and  423.) 

It  may  be  demonstrated  that  the  least  real  radius  of  the 
wheel  should  be  equal  to  the  proportional  radius  added  to 
half  the  pitch;  when  the  necessary  allowances  are  made 
for  wear,  (see  Art.  43.)  and  when  the  staves  are  of  the 
same  diameter  as  the  thickness  of  the  teeth.  But  when 
the  staves  are  larger  than  the  teeth,  as  in  the  figure,  a  less 
real  radius  is  required. 

Having  considered  the  case  of  a  wheel  and  trundle,  with 
cylindric  staves  acting  together,  we  are  now  to  explain  that 
of  a  wheel  and  pinion,  two  sides  of  the  figure  of  whose 
teeth  are  straight  lines  directed  to  its  centre. 

OF  THE  WHEEL  AND  PINION. 

To  find  the  figure  of  the  teeth  and  leaves  of  a  wheel 
and  pinion^  when  thai  part  of  the  teeth  and  leaves^  which 
lies  within  t/ieir  respective  proportioned  circles  are  straight 
lines  directed  to  the  centres  of  these  circles. 

30.  Having  set  off  upon  the  proportional  circles,  the 
points  G,  Q,  L,  and  o,  o,  h,  ice,  according  to  the  thick- 
ness of  the  teeth  and  leaves,  draw  lines  from  these  points, 
tending  towards  the  centre  of  their  respective  circles,  to 
serve  as  the  sides  of  the  spaces  between  the  teeth  and  be- 
tween the  leaves,  the  depth  of  which  spaces  must  be  such 
as  to  give  room  for  the  action  of  the  curved  parts  of  the 
teeth  and  leaves. 


Then  describe  upon  the  extremities  of  the  sides  of  each 
tooth,  epicycloids,  such  as  qd,  ld,  with  the  generating 
circle  t,  the  diameter  of  which  is  equal  to  the  proportional 
radius  of  the  pinion,  upon  the  circumference  of  the  propor- 
tional circle  of  the  wheel  as  a  base. 

The  mode  of  forming  the  teeth  being  thus  shown,  that  of 
the  leaves  will  be'  plain,  v  being  the  generating  circle  of 
their  epicycloid,  upon  the  circumference  of  the  circle  of  the 
proportional  pinion  as  a  base. 

We  have  seen*,  if  the  radius  bh  of  the  proportional 
*  Ch^.  I.  Article  20. 


S6  ON   THE   TEETH   OF   WHEELS.  [CHAP.  U. 

pinion,  be  pushed  by  an  epicycloid  cp,  generated  by  the 
circle  y,  upon  the  pitch  line  of  the  wheel,  and  projecting 
therefrom,  the  pinion  shall  turn  with  the  same  velocity  as 
the  wheeL 

In  the  same  manner  it  may  be  proved,  that  the  same 
eflFect  will  be  produced,  if  the  epicycloid  o  m  m,  attached  to 
the  pinion,  be  pushed  towards  the  line  of  centres,  by  the 
radius,  l  f,  of  the  wheel. 

Lastly,  the  two  opposite  sides  of  the  teeth,  and  those  of 
the  leaves,  ought  to  have  the  same  figure,  for  the  ease  of 
action,  and  to  give  the  wfieel  and  pinion  the  liberty  of  being 
moved  in  either  direction. 

From  these  principles  it  will  be  evident,  that  the  figure 
here  given  to  the  teeth,  will  make  the  wheel  and  pinion 
move  with  perfect  regularity. 

31.  To  make  that  part  of  a  tooth  which  is  within  the 
pitch  line  or  proportional  circle  a  straight  line,  as  proposed 
by  the  Author,  seems  to  be  the  most  advantageous  form, 
because  it  causes  least  pressure  on  the  axes. 

When  the  teeth  are  small,  and  do  not  begin  to  act  till 
they  arrive  at  the  line  of  centres,  the  teeth  of  the  wheel, 
when  the  wheel  drives  the  pinion,  or  the  leaves  of  the  pinion, 
when  the  pinion  drives  the  wheel,  may  be  described  by  a 
circular  arc,  of  which  the  radius  is  equal  to  the  pitch  ;  aod 
of  which  the  centre  is  in  the  pitch  line  of  the  wheel  or 
pinion. 

This  method  will  always  enable  a  workman  to  execute 
short  teeth  nearer  to  the  true  form  than  any  pattern  tooth 
will  enable  him  to  do.  Pattern  teeth  and  compound 
curves,  are  things  that  may  on  some  occasions  be  very 
useful ;  where  the  teeth  are  long,  and  of  considerable  mag- 
nitude in  respect  to  that  of  the  wheel  or  pinion  to  which 
they  belong.  But  in  all  the  ordinary  forms  of  wheel-work 
such  operations  must  consume  an  immense  quantity  of 
valuable  labour  to  attain  even  the  same  degree  of  accuracy 


I   CH,*P.  11-3  ON    THE   TEETH    OF    WHEELS.  27 

that  is  at  once  obtained  by  means  of  circular  arcs.  When 
a  Iiattem  tooth  is  neccssarj',  one  of  its  adjustments  should 
be  the  centre  of  the  wheel ;  and  not  two  points  in  its  cir- 
cumference, as  projrosed  in  Imison'a  Elements  of  Science 
and  Art,  (Vol.  I.  p.  103,)  because  the  latter  method  at 
least  doubles  the  risk  of  error  in  adjusting  the  pattern. 

Wlicn  part  of  the  action  takes  place  before  the  teeth 
arrive  at  the  line  of  centres,  the  method  of  forming  teeth 
proposed  by  our  Author,  (Art.  41,)  seems  to  he  equal,  if 
not  superior,  to  any  other.  And  its  practical  application 
is  shown  in  Art.  41*2. 


pini 
poii 
dra 

.     sbaj 


REMARKS. 

32.  As  it  is  the  curved  part  of  the  teeth  of  the  wheel, 
it  should  push  the  straight  flank  hk,  of  those  of  the 
pinion,  in  removing  from  the  line  of  centres,  and  as  the 
point  E,  where  the  flank  is  acted  upon,  is  a  perpendicular 
drawn  from  a,  it  shall  be  always  that  by  which  the  wheel 
sbaQ  push,  it  is  clear,  that  when  the  extremity-  p,  of  the 

licycloid  c  p,  reaches  the  point  e,  it  shall  cease  to  move 
tooth  H  K  ;  if  the  extremity  p,  arrive  at  the  point  e,  be- 

►rc  the  flank  o  n,  of  the  following  tooth  of  the  pinion  has 
reached  the  line  of  centres,  the  curved  part,  o,  m,  m,  of 
this  tooth,  must  be  pushed  by  the  straight  flank  l  i,  of  the 
following  tooth  of  the  wheel,  till  the  flank  o  n,  reaches  that 
tine :  so  that  in  this  case,  the  wheel  conducts  the  pinion, 
at  one  time  before,  and,  at  another,  beyond  the  line  of 
centres. 

But  wore  it  so,  that  the  extremity  p,  did  not  reach  the 
point  E,  till  after  the  flank,  o  n,  hail  arrived  at  the  line  of 
centres ;  it  would  not  be  necessary,  that  the  curved  parts 
of  the  leaves  should  be  pushed  by  the  flanks  of  the  teeth. 
Thus,  in  this  case,  the  wheel  would  conduct  the  pimon,  in 
pushing  its  leaves  beyond  the  line  of  centres  only. 

3S.  It  is  the  general  opinion  of  those  who  are  in  the 


28  ON   THE  TEETH   OF   WHEELS.  [cHAP.  XI. 

practice  of  constructing  wheel  work,  that  teeth  ought,  if 
possible,  never  to  begin  to  act  before  they  reach  the  line  of 
centres,  as  that  mode  of  action  is  thought  to  occasion  much 
unnecessary  friction*.  The  cause  of  this  great  unneces- 
sary friction,  when  the  teeth  are  of  wood,  appears  to  be  the 
following : 

Friction  depends  not  only  upon  the  pressiure  made  on 
moving  bodies,  but  on  the  inequalities  of  the  surface  upon 
which  they  move ;  and  as  the  surfaces  even  of  the  most 
highly  polished  bodies  have  some  inequalities,  whenever 
two  of  them  are  pressed  together,  the  inequalities  of  the 
one  must  enter  the  other. 

Suppose  A  and  b  to  be  a  wheel  and  pinion,  having  wooden 
teeth,  as  they  would  appear  through  a  microscope ;  it  is 
impossible,  though  there  be  no  other  resistance  than  that 
arising  from  friction,  to  move  them  towards  the  line  of 
centres,  until  either  the  centres,  on  which  the  wheels  turn, 
give  way,  or  some  of  the  small  inequalities,  c,  <f,  of  the 
teeth  be  broken  oflFf . 

On  the  other  hand,  a  very  small  force  will  move  the 
teeth  outwards  from  the  line  of  centres,  as  the  small  ine- 
qualities, c,  d^  and  a,  5,  may  then  slide  over  one  another 
without  being  broken  ;  for  the  teeth,  when  so  working,  are 
mutually  receding  from  each  other  in  their  point  of  contact, 

*  The  increase  of  friction  is  not  the  only  disadTantage  from  part  of  the 
action  taking  place  before  the  teeth  arnTe  at  the  line  of  centres.  For  when 
a  machine  becomes  wonu  it  causes  considerable  irr^ularity  in  its  move- 
mentSy  in  consequence  of  the  action,  in  approaching  the  line  of  centres, 
tending  to  spread  the  axes  of  the  wheels ;  while  the  action,  in  receding  from 
that  lino,  tends  to  draw  these  axes  together ;  and  hence  occasions  more 
inrc^ular  action,  friction^  and  wear  in  the  machine,  in  proportion  to  the  wear 
of  the  parts. 

We  hare  an  obvious  remedy  in  making  pinions  sufficiently  large  to  allow 
of  the  action  being  wholly  on  one  side  of  the  line  of  centres. 

t  See  a  further  iUuslratioQ  of  thb  subject  in  ike  Supplementary  Obscrra- 
tious*    Art*  6$,  &e« 


HAP.  11.] 


ON   THE   TEETH    OP   WHEELS. 


and  the  wheels  move  on  their  centres  with  ease ;  and  where- 
as, in  the  first  case,  they  must  have  a  tendency  to  force  the 
utres  on  which  they  turn  outward,  fi'om  their  true  position, 
I  the  second,  they  have  no  such  tendency*. 


,  when  the  teeth  are  of  metal,  this  unnecessary 

friction  seems  to  arise  principally  after  the  teeth  arc  in 

some  degree  worn.    (See  Fig.  18,  in  p.  36.)     The  teeth  in 

that  case  have  a  kind  of  seat  formed  at  their  bottom,  and 

the  curve  at  the  outward  extremity  is  too  much  inclined  to 

the  radius,  and  very  abrupt.     In  the  action  which  takes 

place  before  the  line  of  centres,  the  eliding  of  the  teeth  of 

;  conducting  wheel  along  those  of  the  conducted,  has  a 

ndency  to  accumulate  hardened  grease,  dust,  sand,  &c., 

(  the  bottom,  which  getting  between  the  abrupt  extremity 

■  the  tooth  and  the  seat  at  the  bottom,  become  like  the 

Ebv-stone  of  an  arch,  and  must  require  often  a  considerable 

fcrce  to  bruise  the  teeth  in  this  situation  past  the  line  of 

ceutres. 


•  Tbc  true  eaiisc  of  tlic  increase  of  frictioti  is  shown  in  Art.  €7.  Tbe 
mnbon  againKt  the  grain  of  lUa  wood  will  ecoicely  have  a  scDsible  eSbct, 
ncept  when  liia  tceUi  aic  new. 


iSm  TBE  TEETH  0&  WHEELS.  [CHAP.  IL 

Tbt»  it  appeanVy  that  the  fridioD  of  teeth,  approaching 
the  line  of  oentreB,  u  modi  greater  dian  in  receding  from 
it  But  in  cases  where  the  pimoo  is  smalls  the  action  in 
approaching  to  the  line  of  centres,  cannot  be  altogether 
prevented.  M.  Camos,  in  his  ^  Conrs  de  Mathematique," 
has  demonstrated,  that  a  wheel  of  50  teeth  cannot  conduct 
a  pinion  of  7  leaves,  without  their  acting  partly  before  they 
arrive  in  the  line  of  centres.  He  also  proves  the  same  with 
regard  to  57  teeth  and  8  leaves,  64  and  9»  7^  su^d  10*. 

34.  Since  the  calculations  which  Camus  has  given  are 

confined  to  particular  cases,  and  troublesome  to  apply  to 

new  ones,  because  his  method  is  indirect ;  we  will,  in  these 

'  additional  articles,  investigate  a  rule,  which  will  be  found 

somewhat  more  general,  and  of  easier  application. 

I^t  B  bo  the  centre  of  the  pinion,  and  c  the  centre  of 
the  wheel ;  then  we  have  to  ascertain  the  relation  between 
the  number  of  teeth  on  the  pinion  and  the  number  of 
those  on  the  wheel,  so  that  any  tooth  d  of  the  wheel  may 
arrive  at  the  line  of  centres  before  the  preceding  tooth  of 
the  wheel  quits  the  tooth  a  of  the  pinion. 

The  last  point  of  contact  cannot  be  beyond  the  middle 
point  of  the  tooth,  and  therefore  the  line  a  c  will  be  in  the 
middle  of  it,  a  being  the  point  of  contact.  Also  ad  will  be 
porpondieular  to  b  a,  the  point  d  being  in  the  pitch  lines. 

Now  by  the  principles  of  trigonometry  we  have  this  pro- 

_A»                              •             •           BC.  sm.  c      Tfc  *    • 
portion;  ab  :  bc  ::  sm.  c  :  sm.  a  = .     But  sm. 

AB 

A  —  sin.  c  =  sin.  b;  therefore,  sin.   c   (—  —  1)  =  sin.  b. 

AB 

The  primitiw  radius  of  a  wheel  or  pinion  being  proportional 
to  the  umuber  of  its  teeth,  it  may  always  be  represented  by 
that  number ;  and  if  n  be  the  number  of  teeth  in  the  wheel. 


CHAP.  11.3  ON   THE   TEETH    OF   WHEELS.  31 

Pio.  B. 


and  n  those  of  the  pinion,  we  shall  have  b  c  =  n  +  n.  The 
line  B  A,  being  a  side  of  a  right-angled  triangle,  considering 
the  radius  unity,  is  found  by  this  proportion;  1  :  cos. 

B  : :  n  :  AB  =  71.  cos.  b.    And  the  angle  b  is  — 


qnently,  sin.  c  (—  —  1)  =  sin.  b  =  sin.  c  (  360       ) 

AB  " ""    


N  +W 
(  t 

n  cos.  - 


35.  As  the  sin.  c  is,  in  practical  cases,  sensibly  equal  to 
its  corresponding  arc,  we  may  use  the  arc  for  the  sine,  which 
will  render  the  equation  much  more ,  simple.    The  arc  is 


a  X  34161      3  _ 47124 


i  which  being  substituted  fbr  sin. 


c,  we  have 


47124  n  (I -cos.—) 


„    ,              .360            360      1    .    2x360. 
Or,  because  sin. —  x  cos.  —  =~-9m. 

(Gregory's  Trigonometry,  Chap.  iv.  Art.  20.) 


32  ON   THE   TEETH   OF   WHEELS.  [CHAP.    II. 

360^ 

9'4248  (1— COS.— ) 

N  = 


.     720     9-4248 
sin. 


n  n 


This  equation  gives  a  result  nearer  to  the  conditions  re- 
quired in  the  actual  construction  of  wheels  and  pinions 
than  when  the  sine  is  used  instead  of  the  arc ;  for  it  makes 
an  allowance  for  the  wearing  away  of  the  teeth.  It  sup- 
poses the  teeth  of  the  pinion  to  he  the  same  size  as  the 
teeth  of  the  wheel,  which  is  the  least  thickness  that  can 
he  given  consistent  with  strength  and  durahility. 

S&.  The  ahove  equation  shows,  that  when  a  pinion  has 
less  than  10  leaves,  it  cannot  he  conducted  uniformly  hy  a 
wheel  with  any  numher  of  teeth  whatever,  unless  they  act 
partly  hefore  they  arrive  at  the  line  of  centres ;  because 
the  denominator  of  the  second  member  of  the  equation  be- 
comes negative  when  n  is  less  than  10. 

This  is  manifest,  for  when  n  =  10,  the  angle  in  the  nu- 
merator is  36  degrees,  and  that  in  the  denominator  is  72. 
But  the  natural  sine  of  72^  as  shown  by  the  Trigonome- 
trical Tables,  is  '95106  when  estimated  in  parts  of  the 
radius,  and  the  second  term  in  the  denominator  when  di- 
vided by  10,  becomes  •94248,  which  being  less  than  the 
sine  of  72**,  the  difference  between  them  is  a  positive  quan- 
tity, and  will  continue  so  for  any  value  of  n  that  is  greater 
than  10;  but  for  any  integral  value  of  n  that  is  less  than 
10,  the  second  term  in  the  denominator  will  exceed  the 
first,  and  consequently,  the  difierence  must  be  negative, 
and  a  negative  number  of  teeth  in  a  wheel  is  obviously 
absurd. 

But  a  pinion  of  10  leaves  may  be  moved  uniformly  by  a 
wheel  with  209  teeth,  when  the  whole  of  the  action  is 
after  they  arrive  at  the  line  of  centres ;  for  in  this  case 


Bap.    II.]  ON    THE   TEETH    OP   WHEELS. 


33 


n=  10,  andbyatableof§^ines,wefindcos.36''  =  -80902;  aud 


8iii.7'2'  =  -95106.  Therefore 


9-4-i2-i-8(i  --soooa) 


=  209  =  N. 


■95106 -■94(<i48 
In  like  manner  it  may  be  calculated  that  a  pinion  of  11 
leaves  may  be  moved  uniformly  by  a  wheel  hav-iog  not  less 
I  than  28  teeth  ;  and  a  pinion  of  1 2  leaves  when  the  wheel  has 
Kit  less  than  15  teeth;  so  that  the  whole  of  the  action 
lay  be  after  the  teeth  arrive  at  the  line  of  centres. 
137.  The  same  equation  may  easily  be  extended  to  the 
5  of  the  wheel  and  trundle,  and  as  it  docs  not  appear  to 
have  been  investigated,  we  shall  here  show  the  result 
under  the  following  conditions  j  1.  The  extremity  of  each 
tooth  of  the  wheel  is  to  be  one-third  of  its  thickness  at  the 
root.  2.  The  centre  of  the  stave  is  to  be  in  the  pitch  line 
when  the  point  of  the  tooth  quits  the  preceding  stave. 
When  limittid  to  these  circumstances  the  equation  in  Art. 


I> 


540         7-33 


Whence  it  appears  that  a  trundle  with  less  than  eight 
staves  cannot  be  moved  uniformly  by  a  wheel  with  any 
Dumber  of  teeth  whatever. 

TTic  same  is  true  also  of  stave-formed  teeth  (see  Art. 
41.)  when  the  wheel  drives  the  pinion, 

38.  From  what  we  have  already  said,  it  will  be  evident, 
when  the  pinion  consists  of  such  a  number  of  teeth,  as  to 
be  conducted  uniformly  by  the  wheel  in  receding  only  from 
the  line  of  centres,  that,  except  in  small  numbers,  the  epi- 
cycloid is  necessary  on  the  conductors  only,  whether  it  be 
a  wheel  or  pinion.  For  instance,  in  Fig.  16,  which  repre- 
sents two  wheels  of  equal  numbers,  a  is  the  conducting, 
and  B  the  conducted  wheel.     But   it  is  to  be  observed. 


I 


34 


ON   THE  TEETH   OF   WHEELS.  [CHAF.    II. 


when  of  two  wheels  acting  on  each  other,  sometimes  the 
one,  and  sometimes  the  other,  is  the  conductor,  the  teeth 
of  both  should  be  epicycloidal,  as  in  Fig.  14.  p.  25. 

When  the  teeth  of  the  conducted  wheel  or  pinion  are 
acted  upon  by  those  of  the  conductor,  in  reccing  only, 
from  the  line  of  centres,  it  may  be  remarked,  if  they  were 
perfectly  made,  and  of  durable  materials,  it  would  be  lu- 
necessary  to  extend  the  conducted  teeth  beyond  their  pro- 
portional circle.  But  these  properties  being  unattainable, 
and  as  the  angles  which  terminate  their  sides,  would  be 
apt  to  cut  the  conducting  teeth,  and  occasion  an  irregular 
motion,  it  is  proper  to  form  the  extremities  of  the  teeth  of 
the  conductor,  in  the  manner  represented  in  the  figure  by 
the  dotted  lines. 


89.  Sometimes  it  may  be  requisite  to  have  but  few  teeth 
in  the  pinion.  In  such  cases,  in  the  conducted,  whether 
wheel  or  pinion,  Buchanan  preferred  staves  to  teeth,  pro- 
perly so  called,  or  to  leaves,  because  a  trundle  or  wheel, 
whose  staves  are  cylindric,  will  be  less  acted  upon  in  ap- 
proaching the  line  of  centres,  and  consequently  have  less 
friction  than  a  pinion  or  wheel,  the  sides  of  whose  teeth 
tend  to  the  centre. 

Thb  will  appear  by  Fig.  17,  which  represents  a  stave,  a. 


HAP.    11.3  ON    THE    TEETH    OF    WHEELS. 


35 


f  a  trundle,  and  a  leaf,  b,  of  a  pinion,  turning  round  on 
•&e  same  centre,  a,  and  a  tooth  adapted  to  each,  turning  on 
a  common  centre,  b.  The  thickness  of  each  of  the  teeth, 
and  the  proportional  circle  of  both  wheels,  are  the  same, 
and  the  proportional  circles  of  the  pinions  are  also  equal, 
and  teeth  are  each  made  of  the  greatest  length  which  the 
intersection  of  the  curves  will  admit,  which  turns  out  con- 
siderably greater  in  the  tooth  adapted  to  the  stave.  The 
shaded  parts  represent  the  tooth  adapted  to,  and  acting 
upon,  the  stave ;  and  the  dotted  lines  represent  the  tooth 
adapted  to,  and  acting  upon,  the  leaf.  The  teeth,  in  both 
cases,  are  represented  as  just  at  the  point  where  they 
would  cease  to  move  the  leaves  or  staves  uniformly ;  and 
it  appears  the  stave  is  conducted  considerably  further  bo- 
yond  the  lino  of  centres  than  the  leaf;  hence  the  stave  will 
be  less  acted  upon  in  approaching  the  line  of  centres. 


■JO.  A  trundle  has  besides  another  considerable  ad>-an- 
tage  over  a  pinion ;    which  is,  that  it  wears  much  more 


36  ON   THE   TEETH   OF   WHEELS.  [cHAP.  II. 

Fio.  18. 


equally.  Every  one  experienced  in  wheel  work  knows, 
that  when  a  pinion  comes  to  be  considerably  worn,  the 
leaves  take  somewhat  of  the  form  represented  at  a,  which 
is  evidently  the  cause  of  a  great  deal  of  unnecessary  fric- 
tion, and  strain  in  a  machine.  Whereas  no  such  thing 
happens  to  the  trundle*.  The  trundle  has,  however,  a  de- 
fect perhaps  as  bad,  if  not  worse,  that  of  weakness.  Its 
staves  being  supported  at  the  ends  only,  arc  not  long  in 
use  before  they  become  quite  unable  to  bear  any  consider- 
able strain,  and  for  this  reason,  it  is  now  in  a  great 
measure  disused  in  machines.  It  however  appears  to  Mr. 
Buchanan,  that  a  wheel  might  be  made,  which  would  com- 
bine the  advantages  of  both  the  pinion  and  the  trundle, 
and  he  accordingly  had  some  wheels  made  on  that  idea, 
and  they  appear  to  answer  every  expectation. 

*  This  is  a  mistake,  as  trundles,  in  consequence  of  the  surfacee  of  contact 
b^ng  Binal],  become  soon  indented  by  pressure,  and  wear  and  cease  to  turn 
round  in  their  sockets. 


CHAP.    11.3  ON    THE   TEETH    OF    WHEELS. 


37 


These  wheels  were  made  of  cast  iron.  Thcv  were  each 
cast  of  one  solid  mass.  The  upper  figure  represents  the 
t'Jge  view,  and  the  lower  the  section  of  one  of  them  j 
irliereby  is  shown  the  maimer  in  which  the  teeth  are  sup- 
[lorted,  like  the  staves  of  a  trundle  at  each  end,  and  like 
the  leaves  of  a  pinion  at  the  roots,  but  so  verj'  thin  there, 
as  to  run  no  risk  of  having  the  eommon  fault  of  pinions 
just  now  noticed.  They  were  difficult  to  mould :  but 
were  they  to  come  more  into  use,  he  had  no  doubt  in- 
genious workmen  would  soon  get  over  this  obstacle*. 


41.  It  is  mentioned  above,  in  cases  where  the  pinion  had 

few  teeth,  that  in  the  conducted,  whether  wheel  or  pinion, 

£tavc8  should  he  preferred ;    but  it    is  obvious,  that  the 

I     method  just  described,  of  making  a  small  trundle  of  cast 

^Hpn,  would  not  apply  to  a  wheel  of  a  great  number  of 

^^Hves.     Nor  is  it  in  that  case  so  necessary,  as  the  greater 

^Tne  number  of  teeth  are,  the  longer  they  will  be  in  losing 

their    proper    figure.      In  such   cases,    therefore,    staves, 

strictly  speaking,  should  not  be  used,  but  teeth  made  so  as 


'  Bj  casting  separate  plntes,  with  indentn 
D  togetber,  &c  jiinton  might  he  made  siillic 
1  frequently  in  crane-work,  wii 
radng  the  wheels  getting  out  c 


0  fit  the  teeth,  and  bolting 
:nt1y  strung  ;  auch  a  method 
re  it  has  the  iniporiMit  wl- 


ON   THE   TEETH   OF   WHEELS. 


[chap. 


to  produce  the  same  effect— that  is,  having  their  acting 
parts  of  the  figiure  of  a  stave. 

What  is  meant  will  be  better  understood  by  inspecting 
the  figure,  where  the  lines  show  the  alteration  necessary 


on  the  tooth  a,  in  order  to  make  it  produce  the  effect  of  a 
stave ;  which  stave  is  represented  by  the  faint  dots.  The 
dotted  lines  on  d,  represent  the  alteration  requisite  to 
adapt  it  to  the  stave,  it  being  necessar}',  as  formerly  proved, 
to  have  it  a  different  epicycloid  from  what  is  required  to 
adapt  it  to  a  tooth,  whose  acting  part  Is  a  straight  line, 
tending  to  the  centre  of  its  proportional  circle. 

42.  Teeth  seem  to  be  very  well  adapted  for  various 
purposes,  when  formed  on  the  principle  recommended  in 
the  preceding  article.  We  therefore  will  endeavour  to  show 
a  simple  method  of  describing  such  teeth. 


CHAP.  II.]  ON    THE    TEETH    OF    WHEELS. 


3U 


lU 

I 


It  most  be  observed  that  the  teeth  to  resemble  staves 
are  [0  be  always  on  the  conducted  wheel  or  pinion  ;  thus 
affonlinp  the  peculiar  advantage  of  the  wheel  and  trundle 
iu  I'itlicr  increasing  or  diminishing  velocity. 


^V  Let  the  teeth  be  divided  as  usual  on  the  pitch  lines,  e  k, 
I  PC;  and  on  the  conducted  wheel  c  describe  circles,  as 
ihough  (here  were  to  be  staves.  Conceive  the  centre  of 
one  of  these  stave  teeth  to  be  in  the  line  of  centres  at  a, 
;ini|  draw  the  line  a  b  joining  the  centres  of  the  stave  teeth, 
riiun  the  radius  a  i,  from  the  centre  a,  will  describe  the 
curved  side  i  c  of  the  tooth  of  the  conductor,  and  the 
cuired  part  fi  a  of  the  conducted  wheel.  And  since  this 
t  is  equal  to  the  pitch  diminished  by  half  the  diame- 
■  the  circlfi  of  the  stave  teeth,  and  the  centres  will 
■ays  be  in  the  pitch  lines  of  the  wheels ;  all  the  other 
I  may  be  easily  described. 

.  The  real  radius,  when  the  wheel  is  the  conductor, 
■  be  vcrj-  easily  calculated  with  sufficient  accuracy  in 


40  ON   THE   TEETH   OF  WHEELS.  [CHAP.   II. 

this  manner :  Xet  b  dhe  drawn  towards  the  centre,  so  that 
dcia  equal  to  ^  of  the  tooth ;  and  make  a  d  perpendicular 

to  bd;  then  we  shall  have  ^/Ai*  —  AflP  =  6rf.  But  a  b 
is  f  of  the  pitch,  when  the  teeth  of  the  pinion  are  of  the 
same  thickness  as  those  of  the  wheel,  and  a  d  is  sensihly 
equal  to  -5^  of  the  pitch ;  therefore  if  p  =  the  pitch, 

p  X  >/  r^  —  jTT  =^bd.:=:  '4714  p.     And  as  in  practice  we 

may  always  regard  b  d  as  the  difference  between  the  real 
and  proportional  radius ;  it  being  only  a  very  small  quan- 
tity in  excess*,  we  have  this  rule : 

The  real  radius  of  a  wheel,  of  the  construction  now  de- 
scribed, should  be  equal  to  the  proportional  radius  added 
to  '47  times  the  pitch :  and  '47  times  the  pitch,  is  very 
little  less  than  half  the  pitch.  But  when  the  teeth  of  the 
pinion  arc  thicker  than  those  of  the  wheel,  the  real  radius 
may  be  less  than  is  given  by  this  rule. 

The  same  rule  applies  to  a  wheel  to  drive  a  trundle. 

This  approximate  method  was  chosen  by  Tredgold  in 
preference  to  a  more  accurate  one,  because  the  result  is 
exhibited  in  those  terms  which  are  most  directly  compar- 
able  with  the  proportions  founded  on  practical  experience. 

When  the  pinion  is  the  conductor,  the  real  radius  should 
be  the  secant  of  the  angle  contained  by  the  pitch,  when 
the  proportional  radius  is  considered  the  radius  of  that 
angle. 

Thus,  c  e  is  the  angle  contained  by  the  pitch,  and  e  d  is 
sensibly  equal  to  the  proportional  radius ;  but  by  the  form 
of  the  teeth  r  e  is  perpendicular  to  e  d,  consequently  r  d  is 
the  secant  to  the  arc  or  angle  c  e ;  and  the  real  radius  re- 
quired to  impel  the  wheel  till  the  succeeding  tooth  begins 
to  act. 

*  The  excess  is  the  difference  between  the  radius  and  the  secant  of  the 
angle  containing  -J^  of  the  pitch. 


^^tbeir  sid 
^V    It  noi 

I 


BAP.    11.3  OS    THE    TEETH    OF   WHEELS.  41 

44.  Hitherto  the  conducting  teeth  have  heen  considered, 
as  being  always  made  as  long  as  the  epicycloidic  tbrm  of 
tlieir  sides  would  admit.     This  however  is  not  always  ne- 

and  in  some  cases  may  be  improper. 

It  now  remains  to  show,  the  smallest  real  radius  a  wheel 
adapted  to  a  trundle  can  have,  without  destroying  the  uni- 
formity of  the  motion. 

\VTien  the  stave  e  (Fig.  13,  p.  22.)  shall  have  been  con- 
ducted to  the  situation  in  which  it  is  represented,  the  point 
S  of  the  following  stave,  shall  be  in  the  line  of  centres  g  f. 
The  stave  a,  in  its  turn,  may  then  be  conducted  by  the 
following  tooth,  t  y  v,  and  then  it  shall  no  longer  be  abso- 
lutely necessary,  that  the  tooth  n  o  s  conducted  the  stave  e. 
The  tooth  k  o  s  may  therefore  be  terminated  in  the  point 
X,  where  it  should  touch  the  stave  e,  when  the  point  t  of 
the  following  stave  shall  be  in  the  line  of  centres,  and  the 
distance  x  f  of  this  touching  point,  from  the  centre  of  the 
wheel,  shall  be  the  least  real  radius  which  can  be  given  to 
the  wheel. 

To  determine  the  point  x,  draw  from  the  centre  of  the 
stave  E  to  the  point  r,  the  straight  line  e  t,  and  where  this 
line  meets  the  circumference  of  the  stave  e,  you  have  the 
point  required*. 

45.  The  smallest  real  radius  which  can  be  given  to  a 
wheel,  adapted  to  the  leaf  of  a  pinion,  or  the  teeth  of  a 
vhccl,  must  evidently  be  terminated  by  that  point  a  of  its 
tooth,  which  is  in  contact  with  the  tooth  or  leaf  e,  after  it 
lias  conducted  it  just  until  the  tooth  following  beirins  to 
act :  thus  ab\%  the  smallest  real  radius  of  the  wheel  c  t. 

*  Sec  Art.  SO.  and  43.,  where  eucL  proportions  as  are  applicable  in 
pi»ctic«  are  ^ven. 
+  Wlicn  the  wheel  isconductor,  if  we  pursue  the  some  mode  of  coIculBtioD 

aa  in  Aft.  43,  making  p=  the  pitch,  we  shall  haver  X  ^/\ = '553  p. 

Tbal  i>,  wben  the  teeth  of  the  conductor  do  not  begin  to  act  till  they  arrive 


42  ON  THE  TEETH  OF  WHEELS.     [CHAP.  H. 

But,  in  practice,  perfect  accuracy  is  not  to  be  expected ; 
and  though  it  were  even  practicable  to  have  wheels  per- 
fectly accurate  when  new,  yet  the  moment  they  are  put  in 
motion  they  begin  to  wear,  and  deviate  from  the  true 
figure  of  their  teeth.  It  would  therefore  be  attended  with 
bad  consequences,  to  make  the  real  radius  no  greater  than 
what  we  have  here  determined  it  to  be,  which  is  the  least 
which  can  be  given.  How  much  greater  it  should  be,  may 
be  determined  by  circiunstances*. 

46.  But  it  appears  to  us,  that  when  wheels  are  made 
with  their  conducting  teeth  only  epicycloidic,  and  the  mo- 
tion is  steady,  there  is  not  much  danger  of  their  being  too 
long  ;  for  the  longer  the  teeth  are,  the  greater  number  of 
them  will  be  in  action  at  the  same  time,  and  consequently 
the  strain  will  be  more  general,  which  will  cause  them  to 
retain  their  true  form  longer.  Besides,  though  a  tooth, 
fit>m  any  accident,  should  be  broken,  the  wheel  will  con- 
tinue to  go  very  well  for  a  long  time ;  whereas,  had  its 
teeth  been  short,  the  wheel  would,  by  such  an  accident, 
have  been  rendered  useless.  We  are  however  aware,  that 
very  long  teeth  are  less  able  to  sustain  any  sudden  stress 
upon  their  extremities  t. 

But  even  supposing  the  teeth,  made  in  the  manner  above 
described,  to  be  no  longer  than  those  formed  epicycloidic, 
upon  both  the  conductor  and  conducted,  yet  the  former 

at  the  line  of  centres,  the  real  radius  should  be  equal  to  "553  times  the 
pitch  added  to  the  proportional  radius.  But  when  the  pinion  is  conductor, 
the  proportional  radius  is  to  the  real  radius,  as  the  radius  to  the  secant  of 
the  arc  equal  to  the  pitch. 

*  Buchanan  had  been  informed,  that  Mr.  Watt  drew  the  figure  of  teeth 
with  segments  and  points,  what  is  below  as  well  as  what  is  above  the  pitch 
lines,  and  that  for  small  strains  and  great  velocities,  he  used  the  pitch  line 
near  the  root  of  the  driver,  but  upon  other  occasions,  he  used  it  just  in  the 
middle,  as  giving  the  greatest  number  of  tovA^hing  points^  which  certainly  adds 
to  the  strength  of  the  wheels. 

t  See  the  last  paragraph  of  Art  70. 


CHAP.  U.]  ON  THE  TEBTR   OF  WHEELS.  43 

will  luTe  less  frictioii,  and  consequently  wear  longer  than 
the  latter,  and  if  th^  be  of  the  same  length,  and  the  same 


diickness  at  the  roots,  they  most  he  equally  strong.     Let 
Fig.  21  represent  wheels  having  the  teeth  of  both  con- 


ductor and  conducted  epicycloidic     In  Fig.  S@,  those  of 
the  OHiductor  only  are  epicycloids,  and  by  inspecting  the 


44  ON   THE   TEETH   OF   WHEELS.  [CHAP.  II. 

figure,  it  will  be  made  obvious,  that  the  teeth  of  a  b,  even 
when  the  wheels  are  new,  must  act  as  much  upon  each 
other,  in  approaching  the  line  of  centres,  as  they  do  in  re- 
ceding from  it.  Whereas  the  teeth  of  c  d,  when  new,  do 
not  begin  to  act  until  they  arrive  in  the  line  of  centres ; 
and  c  conducts  d  much  further  beyond  that  line  than  a 
does  B ;  and  even  when  much  worn,  c  d  acts  but  very  little 
before  the  line  of  centres.  But  it  was  formerly  observed, 
that  when  pinions  have  few  teeth,  they  must  act  before 
they  arrive  in  the  line  of  centres,  and  consequently  in  such 
cases,  the  teeth  of  both  the  conductor  and  conducted, 
ought  to  be  epicycloidic.  Hence  arises  one  of  the  dis- 
advantages of  wheels  and  pinions  having  few  teeth,  a  fault 
carefully  avoided  by  every  good  mechanic. 

It  has  been  mentioned  to  me,  that  the  following  rule,  in 
order  to  determine  the  length  of  the  teeth  of  wheels,  is 
employed  by  the  ingenious  Mr.  Murray  oi Leeds*. 

BULB  TO  DBTBBMINB  THB  LBNOTB  OP  THB  TBBTH  OP  WHBBLS. 

47.  Perpendicular  to  the  line  of  centres  c  d,  draw  the 
line  A  B,  a  tangent  to  the  pitch  lines.  Take  half  the  pitch, 
that  is,  half  the  distance  between  the  centres  of  two  ad- 
joining teeth,  within  a  pair  of  compasses,  setting  their 
points  upon  the  pitch  lines  £  and  f,  parallel  with  the  line 
of  centres  c  d,  draw  the  line  a  6,  and  where  that  is  cut  by 
the  line  a  b  at  c,  gives  the  points  of  the  teeth  of  wheel  and 
pinion  t. 

*  This  we  are  infonned  was  communicated  to  him  by  the  late  Mr.  Rapp, 
of  Manchester,  a  native  of  Germany. 

t  This  rule  is  founded  on  the  properties  of  involute  teeth ;  for  it  may 
easily  be  proved  that  the  line  a  b  will  always  be  divided  in  the  same  ratio  as 
the  pitch  line  divides  the  distance  between  the  centres  of  the  wheels ;  and 
that  the  pitch  line  always  divides  the  length  of  the  teeth  in  that  ratio,  in  in- 
volute teeth.  Hence  the  observations  in  the  following  article  of  the  text, 
are  to  be  understood  as  if  made  on  involute  teeth ;  the  properties  of  which 
our  author  has  not  been  much  acquainted  with.    See  Chap.  III.  Art.  62. 


GBAP.    II.3  OX    THB    TEETH    OF   WHEELS.  4^ 

Fkk  23. 


OBSERVATIONS. 

48.  On  this  rule,  Buchanan  remarks  that  it  does  not 
seem  to  he  founded  on  any  satisfactory  principle ;  were  the 
pinion,  at  all  times,  the  conductor,  he  should  not  perhaps 
differ  ^m  Mr.  Murray,  because  the  action  of  the  teeth 
would  he,  in  that  case,  generally  ojier  their  arrival  at  the 
line  of  centres. 

But  in  case  the  wheel  were  the  conductor,  the  action  of 
the  teeth  would  generally  he  almost  entirely  in  approaching 
the  line  of  centres. 

The  evils  arising  from  this  mode  of  action,  have  already 
been  clearly  proved,  (see  Art.  37,)  it  is  therefore  unneces- 
sary here  to  repeat  them. 

When  the  wheel  and  pinion  are  nearly  of  the  same 


46 


Oir   TBI   TEETH   OF   WHEELS.  £CHAP.  IL 


diameters,  as  in  Fig.  S3,  Uie  effects  are  not  so  obvious  as 
when  the  pinion  is  much  smaller  than  the  wheel,  as  in 
Fig.  24. 


OF  THE  INTERNAL  PINION. 

49.  When  a  pinion  is  to  act  internally,  as  in  Fig.  25,  it 
is  evident,  that  the  teeth  may  be  formed  on  the  principles 
already  laid  down,  with  this  difference  only,  that  the  epicy- 
cloid generated  by  the  proportional  circle  of  the  pinion 
upon  that  of  the  wheel,  should  bo  an  interior  epicycloid. 

The  internal  pinion  may  be  adopted  in  many  cases  with 
advantage,  as  it  has  less  Motion  than  the  external  one*. 

*  A  parallel  motion  upon  this  pnnciple  has  recently  been  erected  at  the 
Buk  of  England. 


CHiP.  U.3  ON   THE   TEETH   OF   WHEELS.  4? 

Vio.  25. 


5a  To  illaatrate  this,  (See  Fig.  26.)  let  a  be  the  pitch 
tine  of  a  wheel,  B  that  of  an  internal  pinion,  and  c  that  of 
an  external  pinion. 


Suppose  the  drde  a  to  he  moved  till  the  point  a  arriTes 
U  b,  and  that  the  points  c  t^  in  the  circles  b  c,  have  both 


48  ON  THB   TEETH   OF   WHEELS.  [CHAP.  lU 

moyed  oyer  a  space  equal  to  a  b.  Now  it  is  evident,  that 
the  distance  from  c  to  6  is  much  less  than  that  fix)m  b  to  d^ 
and  consequently  had  the  circles  moved  one  another  hy 
means  of  teeth,  a  tooth  of  the  interior  circle  b,  in  the  same 
part  of  a  revolution,  would  have  slid  over  a  smaller  part  of 
a  tooth  of  the  circle  a,  than  a  tooth  of  the  exterior  circle  c, 
and  therefore  woidd  have  had  less  velocity.  But  other 
things  heing  equal,  the  less  the  velocity,  the  less  the  fric- 
tion ;  an  interior  pinion  has  consequently  less  friction  than 
an  exterior  one*. 

It  is  upon  this  principle,  that  hovelled  wheels  have  less 
friction  than  external  spur  wheels ;  bevelled  wheels  acting 
in  a  mean  situation  between  external  and  internal  spur 
wheels  t. 

OF  THB  BACK  AND  PINION. 

51.  What  is  called  the  rack  and  pinion,  is  used  for 
various  purposes  in  mechanics ;  as  m  jacks  for  raifiing 
great  weights,  and  for  the  opening  and  shutting  of  sluices. 

The  rack  and  pinion  should  be  made  upon  the  principles 
of  spur  gears ;  with  this  difference  only,  that  in  forming 
the  teeth,  the  cycloid  is,  for  reasons  obvious  from  its  defi- 
nition, used  in  place  of  the  epicycloid  t. 

Doctor  Johnson  gives  this  definition  of  the  cycloid :  "  A 
geometrical  curve,  of  which  the  genesis  may  be  conceived 
by  imagining  a  nail  in  the  circumference  of  a  wheel :  the 
line  which  the  head  of  the  nail  describes  in  the  air,  while 
the  wheel  revolves  in  a  right  line,  is  the  cycloid.'* 

Thus  A  B  c,  is  a  cycloid  generated  by  the  point  a,  in  the 
circle  d,  while  it  revolves  on  the  right  line  a  c  §. 

♦  See  Art  Q5. 

t  The  notion  that  bevelled  wheels  have  less  friction  is  not  correct ;  unless 
in  the  case  where  the  teeth  are  in  the  concave  surface  of  a  cone,  as  in  Fig. 


■J 


t  See  Art  71. 

§  Velocity  makes  no  difference  in  friction. 


CHAP,    n.3  ON   THE   TEETH    OF    WHEELS. 

Pio.  27. 


The  subjoined  figure  represents  the  teeth  of  a  rack  and 
pinion,  formed  in  what  seems  the  best  mode  in  cases  where 
a  great  weight  is  attached  to  the  rack. 


The  leaves  of  the  pinion  are  made  as  long  as  the  curve 
will  admit,  in  order  to  prevent  them  from  beginning  to  act 
before  they  arrive  in  the  Une  passing  through  the  centre  of 
the  pinion,  perpendicular  to  the  rack.  Were  they  to  act 
much  before  they  arrived  in  that  line,  which  may  be  con- 
sidered as  the  line  of  centres,  and  against  a  very  great 
weight,  they  would  be  apt  to  jam,  and  run  the  risk  of  their 
being  broken,  or,  at  least,  very  much  increase  the  friction*. 

5S.  The  construction  above  proposed  for  the  rack  and 
pnioD  not  being  exactly  correct,  I  will  here  endeavour  to 
remedy  that  defect 

If  a  pinion  move  a  rack,  and  the  dotted  line  a  b  be  the 

*  See  thu  Chi^  Art.  33 ;  eleo  the  SupplemenUrj  Observetioiia. 


fitdt  tine  of  the  nek  ;  and  tbe  dotted  ckde  c  ■  Ae  ^tA 
tine  of  the  jmuoa  ;  then  the  cnmd  ade  c  d  «f  Ae  loaA 
of  the  pnioa  fboold  be  an  iuTiJiile  of  a  tinit.    Or,  k  is 


I 
that  curve  which  a  point  in  a  cord  iroakl  describe  aa  the 
pinion,  were  the  pinion  turned  by  drawing  the  cord  con- 
stantly iu  the  direction  a  b  ;  the  cord  being  supposed  to  be 
wound  round  the  pitch  circle  of  the  pinion.  Now  as  b  d 
is  the  part  of  the  cord  which  unwinds  from  the  arc  c  b,  it 
is  obvious  that  the  pitch  lines  more  with  equal  velocities; 
and  the  force  acting  constantly  at  tbe  same  distance  from 
the  centre  of  motion,  the  force  wiU  be  constant,  except 
that  variable  part  which  is  lost  in  ftiction. 

53.  In  order  that  the  teeth  of  the  rack  may  be  durable, 
and  not  liable  to  cut  the  face  of  the  teeth  of  the  pinion, 
they  should  extend  beyond  the  pitch  line  of  the  rack; 
and  the  teeth  of  the  pinion  should  be  of  sufficient  length 
for  one  to  move  the  rack,  till  the  following  tooth  arrives  at 
the  point  b.  To  determine  the  length  that  will  fulfil  the 
latter  condition,  or  what  amounts  to  the  same  thing,  t« 


CHIP.   I[.]  OS    THE    TEETH    OF    WHEELS.  51 

d  thp  real  radius  of  the  pmion,  wc  may  suppose  a  line 
(imivu  from  the  point  r,  to  the  centre  e  of  the  pinion  ;  also 
make  r  b  perpendicular  to  c  e,  and  f  d  porpendiciUar  to 
Then,  by  similar  triangles  wo  have  f  b  :  b  d  : :  B  E 

e  real  radius  —  — But  u  d  is  equal  to  the  pitch, 

FB 

d  p  B  to  five  sixths  of  the  pitch,  therefore  the  real  ra- 

■  diiis=: .    That  is,  the  real  radius   should   be    six 

5 

J  Was  of  the  proportional  radius. 

In  ordinan,-  cases,  the  curved  surfaces  of  the  teeth  of 

the  pinion  may  be  described  from  centres  in  the  pitch  circle, 

with  the  radius  d  b.     And  instead  of  making  the  teeth  of 

the  rack  square  to  the  pitch  line,  they  may  be  described 

hv  (he  same  radius  a  o,  from  points  in  the  pitch  line  of  the 

ratk ;  and  the  ends  of  the  teeth  as  well  as  the  hollows  to 

receive  them,  in  the  pinion  may  be  semicircular.     This 

Diode  of  forming  the  teeth  will  make  them  very  strong 

n'tbout  affecting  the  motion. 

Si.  liVhen  the  rack  impels  the  pinion,  the  curved  face 

of  each  of  the  teeth  of  the  rack,  should  be  a  portion  of  a 

cycloid,   (as  a  a.   Fig.  27,)  and  the  leaves  of  the  pinion 

straight  lines  radiating  from  the  centre  of  the  pinion ;  the 

diameter  of  the  generating  circle  for  describing  the  cy- 

cloidal  teeth  should  be  half  tlie  proportional  diameter  of 

the  pinion. 

SECTION  II. 

OP  BHVBL  GEAR. 

965.   Hitherto  our  inquirj-  has  been  confined  to  what  is 
Bod  spur  gear,  or  the  action  of  wheels  and  pinions  whose 
s  are  parallel  i  we  come  now  to  speak  of  what  is  called 
lel  gear,  or  the  action  of  wheels  of  which  the  axes  are 
to  each  other.      As  we  formerly    regarded  the 
of  gpur  gear,  with  teeth  indefinitely  small,  .ir  the 


52 


ON  THE  TEETH  OF  WHEELS.     [CHAP.  II. 


rolling  of  cylinders  upon  the  surfeice  of  each  other,  we  may 
now  regard  the  action  of  bevel  gear  with  such  teeth,  as  the 
rolling  of  cones  in  a  similar  manner. 

In  order  to  illustrate  this,  let  us  suppose  it  is  required 
to  make  one  wheel  move  another,  the  axes  of  which  are  not 
parallel 

Fio.  29. 


c 


Fio.  30. 


Fio.  31. 


>B 


Fio.  32. 


Let  A  B,  A  0,  be  their  axes,  and  d  e,  e  f,  their  propor- 
tional diameters  or  pitch  lines. 


II.] 


ON    THE    TEETH    OF   WHEELS. 


53 


To  the  point  a,  where  the  axes  intersect,  draw  a  e,  a  f, 
then  DAE,  and  e  a  v,  shall  be  the  outline  of  two 
"Cones*,  which  rolling  the  one  upon  the  surface  of  the 
other,  will  hoth  revolve,  so  that,  like  two  cylinders  with 
their  axes  parallelt,  all  the  corresponding  points  in  each, 
shall  move  in  every  part  of  their  revolution  with  equal 
velocity. 

For,  suppose  any  touching  point  v,  the  diameters  of  the 
cones  at  that  point  shall  bear  exactly  the  same  proportion 
to  one  another,  that  their  bases  do.  The  same  may  be 
said  of  every  other  point  on  their  surfaces,  and  conse- 
quently they  shall  revolve  in  the  same  manner  as  two  cy- 
linders having  their  axes  parallel,  and  the  cones  may  be 
considered  as  bevel  wheels  with  indefinitely  small  teeth. 

But,  in  practice,  we  require  finite  and  sensible  teeth : 
in  bevel  gear,  these  are  made  similar  to  those  of  spur  gear, 
with  this  difference,  that  in  spur  gear,  they  are  parallel ; 
but  in  bevel  gear  they  must,  as  is  evident,  diminish  in 
length  and  thickness,  as  they  approach  the  summit  of  the 
cone. 

The  teeth  may  be  made  of  any  breadth,  according  to 
the  strength  required,  and  they  are  thereby  enabled  to 
overcome  a  much  greater  resistance,  and  work  smoother, 
than  is  possible  for  a  common  face  wheel  and  trundle, 
which,  for  that  reason,  are  now  superseded  by  bevel  gear. 
5(J.  The  epicycloid,  which  gives  the  true  curve  to  the 
teeth  of  bevel  gear,  differs  from  that  used  in  spur  gear,  in 
being-  generated  by  the  rolhng  of  one  cone  upon  the  sur- 
face of  another,  while  their  summits  coincide. 

*  TIloM  cones   we  shall  call   llie  proportioiin]  cones  uF  tliu   wliecl   and 


f  liia  Uthc  observed,  when  Uic 
V^or  of  vDc  of  tlic  COIIC8  becomes  it 
e  30.  the  ttoiut  ji  is  ill  the  san 


certain  inclinations,  tlie 
II  Figure  32 ;  in  others,  i 


54  ON  THE  TEETH   OF  WHEELS.  [CHAF.  IL 

Thus  for  example,  in  the  figure,  the  curve,  a  b  c,  is  de- 
scribed by  a  supposed  style,  fixed  in  the  point  a,  of  the 
circumference  of  the  base  of  the  cone  a  d  e,  while  it  rolls 
upon  the  cone  f  c  a  e. 


The  style  a,  being  always  at  the  same  distance  from  the 
point  E,  where  the  summit  of  the  cone  is  fixed,  all  the  points 
of  the  curve  a  b  c,  shall  be  equidistant  from  the  point  e, 
and  consequently  upon  the  surface  of  a  sphere  which  shall 
have  the  point  e  for  its  centre. 

Hence  the  curve  is  called  a  spherical  epicycloid. 

The  circle  a  g  d  h,  which  in  rolling  describes  the  sphe- 
rical epicycloid,  is  named  the  generating  circle  of  that 
curve ;  and  the  part  a  c,  of  the  circumference  upon  which 
it  rolls,  is  called  the  base  of  the  epicycloid. 

When  the  sphere  is  given  upon  which  the  spherical  epi- 
cycloid is  required  to  be  traced,  and  we  know  the  size  and 
position  of  the  rolling  cone  which  should  generate  this  epi- 
cycloid, it  will  be  easy,  from  what  has  been  said  relative  to 
the  plane  epic}'cloid*,  to  find  as  many  points  of  the  curve, 
as  may  be  neccssai}',  and  it  will  be  evident,  that  what  is 
said  on  spur  gear,  respecting  the  most  advantageous  figure 
of  their  teeth,  is  all  appUcable  to  bevel  gear ;  vrith  this 

•  Sec  Chap.  I.  Art.  17. 


■.  u.] 


ON    THE    TEETH    OF    WHEELS. 


fi'renec,  that  the  sphcricaJ  is  substituted  for  the  plane 
epicycloid. 

lu  iirder  therefore  to  avoid  tedious  repetitions,  we  shall 
immediately  proeccd  to  give  some  account  of  what  spems 
I  tho  best  practical  method  of  laying  down  the  lines  nocos- 
sary  to  the  right  construction  of  bevel  gear. 

57.  Ha\ing  calculated  the  projKirtional  diameters  or 
pilth  lines  of  the  wheel  and  pinion,  draw  their  axea,  a  b, 
A  c,  in  the  proposed  direction  with  respect  to  each  other. 


lien  the  wlicels  are  in  action.    Parallel  to  a  b,  and  at  the 

iBtance  of  half  the  proportional  diameter  of  the  wheel, 
draw  the  line  n  e.  In  the  same  manner,  draw  f  d  at  the 
distance  of  half  the  proportional  diameter  of  the  pinion 
from  A  c.  From  the  point  d,  where  these  lines  intersect, 
draw  the  line  d  g,  perpendicular  to  a  b,  and  also  the  line 
11  H,  perpendicular  to  c  a.  Make  g  i  equal  to  i  d,  and  k  h 
equa)  to  k  d.      Then  u  o  is  what  we  shall  call  the  priti- 

pai  diameter,  or  the  dinmeter  at  the  pitch  line  of  the 
«!,  and  D  II  that  of  the  pinion. 
\  Join  G  a,  u  a,  u  a.     Then  o  a  d,  is  the  outline  of  the 

»portional  cone    of  the  wheel,  and  n  a  h,  that   of  the 


L  Now  proceed  to  draw  the  teeth  of  ihe  wheel.     With  the 


56 


ON  THE  T^TH  OF  WHEELS.  [CHAP.  H. 


distance  a  a,  from  a,  as  a  centre,  sweep  a  smaU  arc>  sxich 
as  G  a;  at  the  priDcipal  diameter  to  their  extremity,  set  off 
the  length  of  the  teeth,  from  otob,  and  draw  b  c  tending  to  a. 
The  line  h  c  represents  the  breadth  of  the  teeth,  which, 
according  to  circumstances,  may  be  more  or  less ;  only  it 
is  to  be  observed,  that  if  continued  to  the  point  a,  the 
teeth  near  that  point  would  be  so  small  as  to  be  of  little  or 
no  use. 

Describe  the  arc  c  e,  concentric  to  i  a* ;  and  from  o  to 

Ji  set  off  part  of  the  required  length  of  the  tooth,  from  the 
principal  diameter  to  the  root:  then  dra.vij'g  tending  to 
A,  the  line/g-  becomes  the  root  of  the  tooth.     Parallel  to 

yg,  draw  ae,  then  a,fge,  represent  the  section  of  the 
solid  ring  of  the  wheel.  The  particular  direction  of  the 
line  a  e  is  no  way  essential ;  all  that  is  necessary  is,  that 
the  ring  be  of  sufficient  strength  for  the  purpose  to  which 
it  is  to  be  applied  :  but  patterns  for  cast  iron  wheels  are 
usually  made  as  represented  in  the  plate. 

Fig.  35. 


'  In  practice,  it  is  found  easier,  tuid  sufficiently  accurate,  to  use,  instead 
or  these  curtes,  stnugbt  lines,  as  near  as  may  be,  in  the  same  direction  with 


caAP.  II.J  ON   THE   TEETH    OF    WHEELS. 

Flo.  36. 


I 

^H  Haii-ing  thus  drami  a  section  of  a  tooth  at  g,  draw  in 
^B^  Ae  same  manner  one  at  d,  then  d,  i,  g,  l,  will  be  the  sec- 
tion of  the  wheel,  in  which  e,  h,  i,  l,  a,  represent  the  space 
oaupied  by  the  arms.  The  dimensions  of  these,  and  their 
particular  form,  may  however  be  varied  according  to 
fircumstances. 

The  mode  of  drawing  the  section  of  this  pinion  will  now 
be  obvious,  by  inspecting  the  figure,  where  it  will  be  ob- 
fierred,  that  the  teeth  of  the  pinion  are  made  a  little 
broader  than  those  of  the  wheel.  This  is  a  practice  gene- 
rally followed,  as  the  teeth  by  this  means  wear  more 
i-qually  than  otherwise  they  would. 

5S.  For  the  use  of  young  mechanics,  I  will,  in  these  ad- 
^^Udons,  attempt  to  free  the  principles  of  constructing  be- 
^^blle<l  wheels  of  part  of  their  intricacy,  first  briefly  noticing 
^Hte  old  principles. 

The  teeth  of  bevelled  wheels,  for  moving  one  another 
uniformly,  may  be  formed  according  to  different  principles. 
"hose  which  have  been  delivered,  are,  first,  When  the 
I  drives  the  pinion,  the  acting  faces  of  the  teeth  of 
3  wheel,  should  be  portions  of  a  spherical  epicycloid, 
nerated  by  the  revolution  of  a  cone,  (as  described  in  Art. 


58  ON   THE   TEETH    OF  WHEELS.  [CRAP.  II. 

59))  of  which  the  base  is  half  the  diameter  of  the 
pinion,  and  the  teeth  of  the  pinion  plane  surfaces  directed 
to  its  centre.  (Camus  on  the  Teeth  of  Wheels,  Art  569. 
Brewster,  Edin.  Ency.  voL  xiii.  p.  575.)  Second,  If  the 
teeth  of  the  pinion  be  staves,  or  formed  to  act  as  stay^es, 
then,  when  the  wheel  drives  the  pinion,  the  acting  faces 
of  the  teeth  of  the  wheel  should  be  portions  of  a  curve  pa- 
rallel to  a  spherical  epicycloid,  generated  by  the  revolution 
of  a  cone,  of  which  the  base  is  equal  to  the  diameter  of 
the  pinion.  (Camus,  Art  560.  Brewster,  Edin.  Ency.  vol. 
xiii.  p.  575.)  To  these  a  third  may  be  added,  that  is. 
When  the  pinion  drives  the  wheel,  the  most  advantageous 
form  for  the  acting  faces  of  the  teeth  of  the  pinion,  will 
be  a  spherical  involute  of  a  circle,  the  teeth  of  the  wheel 
being  plane  surfeu^s  directed  to  its  axis. 

The  description  of  these  curves  is  not  a  very  simple 
operation,  nor  yet  adapted  for  application  .in  practice ;  I 
shall  therefore  propose  a  new  method,  which  appears  to 
have  escaped  the  notice  of  former  inquirers,  one  which  is 
general ;  spur  wheels,  racks,  &c.  being  particular  cases  of 
its  application.     See  Fig.  Ej  p.  59.) 

59.  If  A  B  be  the  axis  of  a  pinion,  and  a  c  the  axis  of 
the  wheel,  d  e  the  proportional  diameter  of  pinion,  and  d  f 
that  of  the  wheel,  if  c  b  be  made  perpendicular  to  a  d  ; 
then,  the  proper  form  for  the  acting  surfaces  of  the  teeth, 
may  be  described  upon  the  surfaces  of  the  cones  deb,  and 
D  F  c.  And  since  the  surfaces  of  these  cones  can  be  spread 
out,  or  developed  upon  a  plane  surface,  the  form  of  the 
teeth  proper  for  communicating  equable  motion,  may  be 
drawn  upon  a  plane. 

60.  But  the  developement  of  a  cone  is  a  sector  of  a 
circle  of  which  the  radius  is  the  slant  height  of  the  cone, 
and  the  arc  equal  to  the  circumference  of  the  base  of  the 
cone.  Thus,  if  the  arc  d  g,  be  described  with  the  radius 
c  D  i  this  arc  d  g  will  be  the  developement  of  part  of  the 


CHAP.    II.]  ON    THE    TEETH    OF   WHEELS, 


59 


circumference  of  the  proportional  circle  or  pitch  line,  of 
which  the  diameter  is  f  d,  then  d  c  g  is  part  of  the  deve- 
lopement  of  the  cone,  and  in  the  same  manner  the  deve- 
lopement  dbh,  of  part  of  the  cone  dbe,  may  be  de- 
scribed. For  in  practice,  a  small  portion  of  the  develope- 
ment  is  sufficient. 

Now,  if  the  sectors  that  would  cover  part  of  the  cones 
DBE,  and  DCF,  be  considered  portions  of  spur  wheels, 
and  the  teeth  be  formed,  so  that  these  sectors  would  move 
one  another  equably,  bv  the  methods  described  for  spur 
wheels,  (see  Art,  29,  31,  4.2,  G3,)  on  sheet  copper,  or  any 
other  flexible  body  that  could  be  applied  upon  the  sur- 
faces of  the  cones,  the  outline  of  the  teeth  formed  by 
these  patterns,  would  be  such  as  are  adapted  for  bevelled 
wheels. 

61.  The  breadth  of  the  teeth  do,  being  settled  by  the 


60  ON   THE   TEETH   OF  WHEELS.  (^CHAP.   II. 

parallel  to  c  b.  Then  the  form  of  the  interior  end  of  the 
teeth  may  be  found  by  spreading  out  the  cones  dcf^  and 
dhe.  But  this  is  more  easily  done  by  making  d k  parallel 
to  the  axis  a  c,  and  from  the  point  k^  where  this  line  cuts 
c  B,  with  the  radius  c  Ar,  describe  the  arc  kg^  which  is  part 
of  the  developement  of  the  proportional  circle  at  d.  Now 
lines  drawn  from  the  teeth  on  the  arc  d,  to  the  centre  c, 
will  determine  the  magnitude  of  the  teeth  at  the  arc  k ; 
and  the  teeth  may  be  described  there  accordingly,  as  is 
shown  in  the  figure.  Also,  when  di  \b  drawn  parallel  to 
A  B,  we  shall  have  b  i  equal  the  radius  of  the  developement 
of  an  arc  of  the  proportional  circle  of  the  pinion  at  d  ;  and 
on  this  arc  the  pattern  teeth  should  be  described  for  the 
interior  ends  of  the  teeth  of  the  pinion,  as  indicated  by  the 
lines  in  the  figure. 

Thin  copper  will,  be  very  well  adapted  for  the  patterns, 
and  it  will  be  desirable  that  there  should  be  two  teeth 
upon  each  pattern,  but  more  will  not  be  necessary. 

The  limit  of  the  pitch  will  be  the  same  as  for  spur 
wheels,  (see  Art.  138,)  and  the  breadth  of  the  teeth  is 
also  to  be  regulated  by  the  same  rules. 

The  length  of  the  teeth,  the  friction  of  them,  and  the 
peculiar  advantages  of  the  diflferent  modes  of  forming  them, 
may  be  considered  on  the  developed  pitch  lines  in  the  same 
manner  as  if  they  were  the  pitch  lines  of  spur  wheels ;  con- 
sequently every  remark  that  applies  to  the  one,  applies  to 
the*  other.  Indeed,  the  only  difficulty  in  this  construction 
of  the  teeth  of  bevelled  wheels,  consists  in  applying  the 
patterns  correctly  to  the  conic  surface  whereon  the  ends  of 
the  teeth  are  to  be  described ;  but  it  is  a  difficulty  which 
is  very  easily  overcome  by  having  proper  lines  on  that  sur- 
face, to  adjust  the  corresponding  lines  on  the  pattern  by. 

Perhaps  it  will  be  of  use  to  make  a  few  small  models  of 
wheels,  in  order  to  fully  understand  the  process,  and  apply 
it  with  certainty  of  success.     For  though  it  is  extremely 


CHAP,  n.3  ON  THE  TEETH   OF  WHEELS.  6I 

simple  to  those  accustomed  to  work  by  developed  patterns, 
such  as  carpenters,  joiners,  and  masons*,  it  may  not  ap- 
pear  so,  at  first,  to  others. 

*  See  the  Art  Joinsbt,  Supplement  to  ihe  Encyclopaedia  Britannica, 
{.  11—17;  or  Nicholsons  Carpenter's  Guide,  p.  18. 


6s  ON   THE   TEETH   OF   WHEELS.  [CHAP.  HI. 


CHAPTER  IIL 

62.  We  shall  now  proceed  to  describe  a  mode  of  forming  the 
teeth  of  spur  wheels,  the  first  hint  of  which,  we  have  been 
informed,  was  given  by  Professor  Robison,  of  Edinburgh. 
In  order  to  understand  the  description  and  demonstration, 
it  will  be  necessary  to  recollect  what  was  proved  in  Chap- 
ter I.  Art.  11,  viz.:  That  if  a  wheel  move  uniformly,  it  is 
necessary,  in  order  to  move  another  wheel  uniformly,  that 
the  form  of  the  teeth  be  such  as  that  the  perpendicular 
from  the  touching  surfaces  in  all  situaiions^  cut  the  line  of 
centres,  in  the  same  point  a,  which  point  divides  the  line 
of  centres,  so  that  the  one  part  a  b,  shall  be  to  the  other 
A  G,  as  the  number  of  teeth  in  the  one  wheel  b,  is  to  the 
number  of  teeth  in  the  other  g. 

This  being  understood,  let  b  and  g  be  the  centres  of  the 
two  wheels,  and  def,  ghij  the  rings  upon  which  the  teeth 
are  placed ;  the  diameters  of  which  rings  are  to  one  an- 
other,  as  their  number  of  teeth.  If  the  thread  kd^h^ 
lapped  round  the  ring ;  as  it  folds  up,  its  extremity  Ar,  will 
describe  the  curve  klm.  It  is  evident,  that  the  thread  in 
describing  the  curve,  is  perpendicular  to  it ;  and  the  thread, 
^,  Ijf^  A,  is  therefore  perpendicular  to  the  curve  at  /,  or  a. 
In  the  same  manner  it  may  be  shown,  that  the  point  n,  of 
the  thread  g  n^  will  describe  a  similar  (5urve,  nop^  which 
is  perpendicular  to  the  thread  gn^  hOj  i  a,  at  the  points  w, 
0  and  A.  If  therefore  m,  a,  Z,  Ar,  be  a  curve  formed  by  the 
evolution  of  the  ring  b,  and  nop^he  a  curve  formed  by 
the  evolution  of  the  ring  of  the  wheel  o,  a  line  drawn 
through  the  point  of  contact  a,  perpendicular  to  the  touch- 
ing  surfaces,  will  touch  both  rings  in  the  points  f  and  f. 


CHAP,  in.]  ON    THE   TEETH    OF    WHEELS. 


63 


w^K  ad  the  two  curres,  supposing;  thcra  teeth,  will  act  as  if 
^Bllieone  pxilled  the  other  by  the  thread  /  /.'     The  line  ij". 


wtD  be  the  line  of  action ;  that  is,  the  teeth  will  always 
touch  each  other  in  a  point  of  this  line,  and  since  this  line 
alwATS  passes  through  the  same  point  of  the  line  of  centres, 
H  G,  the  actinn  will  be  invariable  ;  so  that  if  the  one  move 
onifonnly,  the  other  will  also  move  unifonulj",  and  two 
wMghts  which  balance  in  anj/  one  position,  will  balance  in 
aU  positions  of  the  teeth. 

It  is  obvious,  that  though  these  teeth  must  work,  both 
before  and  after  passing  the  line  of  centres,  that  they  will 
work  with  equal  truth,  whether  pitched  deep  or  shallow  ; 
a  quality  peculiar  to  them,  and  of  ver\'  great  importance. 

(i3.  The  following  properties  of  involute  teeth  arc  of 
most  iniportane«  in  wheclwork. 

I.  When  I  he  wheel  drives  the  pinion,  the  greater  part 


64  ON  THE  TEETH  OF  WHEELS.     [CHAP.  III. 

of  the  action  will  take  place  before  the  teeth  arrive  at  the 
line  of  centres. 

2.  When  the  pinion  drives  the  wheel,  they  act  chiefly 
after  passing  the  line  of  centres;  and  therefore  involute 
teeth  are  most  adapted  for  this  case. 

3.  When  the  teeth  are  long,  the  action  is  very  oblique, 
causing  much  unnecessary  stress  upon  the  axes ;  particu- 
larly when  the  wheel  impels  the  pinion. 

It  may  be  remarked,  that  when  a  thread  winds  ofi^  from 
one  circle  to  another,  and  these  circles  touch  one  another 
at  the  circumference,  a  point  in  the  thread  will  describe  an 
epicycloid ;  and  the  same  epicycloid  would  be  described  by 
making  the  circle  from  which  the  thread  winds  ofi^,  the 
generating  circle,  the  other  being  the  base. 

The  length  of  involute  teeth  may  be  determined  with 
sudfficient  accuracy  by  the  rule,  Art.  47- 

SUPPLEMENTARY  OBSERVATIONS. 

64.  The  foregoing  Essay  was  written  several  years  be- 
fore the  publication  of  the  Supplement  to  the  Encyclopa^a 
Britannica.  Professor  Robison  has  there  (VoL  II.  page 
103,  106)  described  and  recommended  the  mode  which 
will  be  found,  Chap.  III.  of  forming  teeth  of  wheels  by 
involutes  of  circles.  Dr.  Brewster,  however,  in  his  second 
Edition  of  Ferguson's  Lectures,  VoL  II.  page  227,  ob- 
serves, that  this  principle  is  not  new ;  De  la  Hire  having 
long  ago  considered  the  involute  of  a  circle,  as  the  last  of 
the  exterior  epicycloids ;  which  it  may  be  proved  to  be,  if 
we  consider  the  generating  straight  line  as  a  curve  of  in- 
finite radius*. 

Professor  Robison  sayst,  that  ^*  this  form  of  teeth  ad- 

♦  See  Art  71. 

t  Encyclop»dia  Britannica,  Volume  XX.  page  104.    See  also  Rees's 
Cydopcedia,  Art.  Clock  Movement 


Bap.  111.]  ON    THE    TEETH    OF    WHEELS.  t)5 

mits  of  several  teeth  to  be  acting  at  the  same  time,  (twice 
the  number  that  can  be  admitted  in  M.  De  la  Hire's  me- 
thod.) This,  by  dividing  the  pressure  among  several  teeth, 
diminishes  its  quantity  on  any  one  of  them,  and  therefore 
diminishes  the  dents  or  impressions  which  they  unavoid- 
ably make  on  each  other.  It  is  not  altogether  free  from 
sliding  and  friction,  but  the  whole  of  it  can  hardly  be 
said  to  be  sensible.  The  whole  slide  of  a  tooth,  three 
inches  long,  belonging  to  a  wheel  of  ten  feet  diameter, 
does  Dot  amount  to  one  sixtieth  of  an  inch,  a  quantity  al- 
together insignificant'. 

In  the  same  article,  this  highly  respectable  philosopher 
was  mistaken,  in  supposing,  with  other  eminent  authors, 
that  the  mutunl  action  of  the  teeth,  (when  formed  into  epi- 
cycloids, by  the  method  of  M.  Camus,)  is  absolutely  witli- 
ouijriction,  and  in  saying,  "  That  one  tooth  only  applies 
iUetfto  the  other,  and  rolls  on  it,  but  does  not  slide  or 
KUB  an  it  in  the  smallest  degree,  TTiis  makes  them,  last 
long,  or  rather  does  not  allmo  them  to  wear"  A  very 
slight  examination  of  the  figures  given  in  various  parts  of 
the  preceding  Essay,  will,  I  hope,  show,  that  the  point  of 
contact  must  slide  from  the  pitch  line  of  the  conducting 
h  outwards.  Dr.  Young,  in  his  Natural  Philosophy, 
•Vol,  II.  page  183,  says,  that  **  a  form  [of  teeth]  without 
JHction,  is  perfectly  impracltcable,  although,  fjr  a  single 
tootht  possible." 

65.  In  the  first  volume  of  the  same  work,  he  makes  the 
foUofring  judicious  observations  on  our  present  subject : 

"  It  baa  been  supposed  by  some  of  the  best  authors  that 
the  epicycloida!  tooth  has  also  the  advantage  of  completely 
aToiding  friction ;  this  is  however  by  jio  means  true,  and 
it  is  even  impracticable  to  invent  any  form  for  the  teeth  of 
a  wheel,  which  will  enable  them  to  act  on  other  teeth 
ithout  friction. 

•  Sm  Art.  70. 


cont 
Lloot; 
fVo\. 


66  ON   THE   TEETH    OF   WHEELS.  []CHAP.  III. 

^^  In  order  to  diminish  it  as  much  as  possible,  the  teeth 
must  be  as  small  and  as  numerous  as  is  consistent  with 
strength  and  durability ;  for  the  effect  of  fidction  always 
increases  with  the  distance  of  the  point  of  contact  from  the 
line  joining  the  centres  of  the  wheels. 

"  In  calculating  the  quantity  of  the  friction,  the  velo- 
city with  which  the  parts  slide  over  each  other  has  gene- 
rally been  taken  for  its  measure :  this  is  a  slight  inaccuracy 
of  conception,  for,  as  we  have  already  seen,  the  actual  re- 
sistance is  not  at  all  increased  by  increasing  the  relative 
velocity ;  but  the  effect  of  that  resistance,  in  retarding  the 
motion  of  the  wheels,  may  be  shown,  from  the  general 
laws  of  mechanics,  to  be  proportional  to  the  relative  velo- 
city thus  ascertained.  When  it  is  possible  to  make  one 
wheel  act  on  teeth  fixed  in  the  concave  surface  of  another, 
the  friction  may  be  thus  diminished  in  the  proportion  of 
the  difference  of  the  diameters  to  their  sum. 

^*  If  the  face  of  the  teeth,  where  they  are  in  contact,  is 
too  much  inclined  to  the  radius,  their  mutual  fricticm  is 
not  much  affected,  but  a  great  pressure  on  their  axes  is 
produced ;  and  this  occasions  a  strain  on  the  machinery, 
as  well  as  an  increase  of  the  friction  on  the  axes.'^  * 

The  concluding  part  of  these  observations  appears  to  me 
peculiarly  applicable  to  the  figure  of  teeth  described  in  our 
last  chapter ;  for  in  wearing,  they  will  be  more  liable  than 
many  other  forms,  to  have  the  face  of  the  teethf  where  tihey 
are  in  contact^  too  much  inclined  to  the  radius. 

REMARKS  ON  THE  FRICTION  OF  WHEEL  WORK,  AND  ON 
THE  FORMS  BEST  SUITED  FOR  TEETH, 

IN  A  LBTTBB  FBOM  DB.   TOUNO. 

'66.  **  I  have  been  considering   your    observations  on 
the  difiereuce  of  the  friction,  accordingly  as  the  teeth 

"*  Youngs  LccUiKs,  VoL  I.  p.  176. 


.  in.] 


ON    THE    TEETH    OF   WHEELS. 


fi7 


» 


y 

r 


bmch  before  or  after  the  line  of  centres ;  at  first  I  was 
disposed  to  doubt  of  the  fact ;  but,  upon  more  mature 
examination,  I  found  that,  like  many  other  practical  oh- 

tservations,  they  went  beyond  the  scope  of  the  doctrines  of 
theoretical  writers.  1  cannot  however  perfectly  agree  with 
you  as  to  the  explanation  of  the  fact ;  but  I  will  state  to 
you  briefly  my  opinion  on  the  subject,  not  having  leisure 
at  present  to  enter  into  a  more  ample  discussion. 

"  The  magnitude  of  the  friction  has  usually  been  esti- 
mated by  the  relative  velocity  of  the  surfaces  concerned  ; 
a  mode  of  calculation,  which,  as  I  have  observed  in  ray 
lecture  on  machinery,  is  only  so  far  correct,  as  it  shows  the 
comparative  effect  of  a  given  friction  in  retarding  the  ma- 
chine. But  in  fact  the  primitive  friction  itself  is  liable  to 
variation,  according  to  the  obliquity  of  the  surfaces ;  for 
since  the  friction  is  nearly  proportional  to  the  mutual  pros- 
sure,  it  will  he  greater  or  less,  as  the  direction  of  these 
Borfaccs  is  more  or  less  inclined  to  the  radii,  the  force  of 
rotation  being  supposed  to  he  given :  and,  what  Is  of  still 
more  immediate  importance  to  the  resolution  of  the  diffi- 
culty  in  question,  the  direction  of  the  force,  by  which  the 
one  wheel  acts  on  the  other,  is  not  to  be  considered  as  per. 

.fendicuiar  to  the  surface  of  the  teeth,  but  as  oblique  to  it, 

%eing  8o  situated  as  to  oppose  the  joint  residt  of  the  direct 
resiatance  and  the  friction  j  that  is,  as  being  inclined  to  the 
Burfaee  in  a  certain  constant  angle,  which  a  late  anonj-mous 
writer  has  called  the  angle  of  repose,  and  which  is  equal 

|to  the  inclination  of  a  plane,  on  which  one  of  the  sub. 

hitances  concerned  would  begin  to  slide  on  the  other  by  its 
grantatiim. 

67.  "  Let  the  tooth  a  impel  the  tooth  b  with  the  given 
fiirce  A  c,  perpendicular  to  the  common  surface  of  the 
;  make  cad  equal  to  the  angle  of  repose,  then  the 
mu^t  act  in  the  direction  a  d,  and  making  c  d  parallel 
the  radius  a  e,  ad,  wilt  he  the  actual  pressure :  then 


68  ON  THE  TEETH  OF  WHEELS.     [CHAP*  IH. 


Fio.  38. 


drawing  d  f  parallel  to  the  radius  a  6,  a  f  will  be  the  effec* 
tive  force  in  the  direction  a  c,  and  f  c  will  be  the  loss  by 
friction.  Again,  if  b  impel  a,  the  angle  of  repose  must  lie 
on  the  other  side  of  a  c,  and  c  h  must  be  parallel  to  a  g, 
and  H  I  to  AE,  and  the  friction  in  this  case  will  be  i  c, 
which  is  obviously  less  than  f  c. 

"  Hence  we  may  easily  calculate  the  magnitude  of  the 
resistance  f  c,  or  i  c,  produced  by  friction,  calling  the  force 

A  c  unity ;  for  c  d  becomes  -^ ,  and  f  c  =  c  d.    -I = 

S.  ADC  S.  CFD 

5.  CAD  S.  CDF      S.  CAD       S.  GAE  I     j  •   xi. 

'  and  m  the  same 


S.  ADC  S.  CFD  S.  (CAE  -f  CAD)  S.  GAC 

_  ^.CAH   5.  CHI      ^.CAD       ^.GAE    tj  .1. 

manner  ic  = . =— ^ n  . .     Both 

^.AHC      S.CIH      5.  (G  AC  +  CAD  J      ^.EAC 

these  quantities  vary  ultimately  as  the  angle  formed  by  the 
radii,  and  vanish  when  the  point  of  contact  is  in  the  line 
of  the  centres ;  and  in  this  case  the  common  theory  agrees 
with  this  calculation*  When  c  a  e  is  always  a  right  angle, 
as  in  the  epicycloidal  tooth  commonly  reconmiended,  the 


CHAP.  Itl.3  ON    THE    TEETH    OF    WHEELS. 


69 


friction  F  c  varies,  in  the  different  positions  of  the  teeth,  as 
-1— : — ,  or  as  cdt.  o  a  c ;    that  is,  if  .\  k  be  made  constant, 

(.  GAC 
as  K  L. 

68.  *'  Since  therefore  it  is  demonstrable,  that  the  friction 
or  pressure  is  always  greater  in  approaching  the  line  of  the 
centres,  than  at  an  equal  distance  beyond  it,  it  must  ob- 
viously be  desirable  that  the  contact  should  be  rather  after 
Ihan  before  the  passage  of  the  teeth  over  that  line,  although 
it  is  better  that  it  should  be  at  a  small  distance  before, 
than  at  a  much  greater  distance  beyond  it.  Hence  the 
impelling  teeth  ought  to  be  of  such  a  form,  as  to  accelerate 
the  motion  of  the  impelled  a  little  before,  and  a  little  more 
after,  the  passage  of  the  line  of  the  centres,  and  then  to 
retard  it  again,  so  that  the  next  tooth  may  succeed  to  a 
similar  operation. 

69.  **  A  wheel  acting  on  a  trundle,  with  cylindrical 
staves,  has  in  this  respect  an  advantage  over  two  wheels 
with  teeth,  since  the  curve,  fitted  for  impelling  the  trundle, 
\a  adapted  only  to  act  on  it  beyond  the  line  of  the  centres. 
This  curve  may  however  be  formed  more  easily,  and  at  the 
same  time  more  advantageously,  than  by  the  method  which 
has  hitherto  been  recommended :  for  if  we  employ  an  epi- 
cycloid described  by  the  rolling  of  a  circle,  which  would 
just  touch  the  internal  surface  of  all  the  staves  of  the 
trundle,  on  the  circumference  of  the  wheel,  the  trundle 
irill  at  first  be  accelerated  a  very  little,  and  will  then  bo 
allowed  to  fall  back  from  each  tooth  to  the  succeeding  one, 
soon  after  its  passage  over  the  line  of  the  centres.  The  same 
form  will  also  answer  verj'  well,  when  the  trundle  is  to  im- 
pel the  wheel,  although  this  mode  of  action  produces  a 

_  greater  friction  than  the  former. 

^K  70>  *'  A  similar  advantage  may  be  obtained  in  teeth  of 
H|Dy  other  form,  by  finishing  them  in  such  a  manner  as  to 
^hrqject  a  very  little  beyond  the  regular  outline,  at  the  point 


70  ON   THE   TEETH   OF  WHEELS*  [CHAP.  UU 

which  is  intended  to  come  into  contact  a  little  beycmd  the 
line  of  the  centres.  Such  a  corrected  outline  may  he  do- 
scrihed  at  once,  if  it  he  required.  If  the  tooth  is  to  he 
formed  into  an  involute  of  a  circle,  having  fitted  a  thread 
or  fine  wire  to  the  circumference  of  the  wheel,  find  the 
point  of  contact  at  the  instant  when  the  end  of  the  wire  is 
describing  the  part  of  the  tooth  which  is  to  act  at,  or  a 
little  before,  the  line  of  the  centres ;  cut  off  fix)m  the  wheel, 
beyond  this  point,  an  arc  equal  to  the  distance  of  the 
centres  of  two  adjoining  teeth,  and  fix  a  pin  in  the  tangent 
at  the  same  point,  that  is,  in  the  continuation  of  the  part 
of  the  wire  which  is  unrolled,  at  such  a  distance  as  just  to 
streteh  the  part  which  is  left  loose  by  the  removal  of  the 
arc :  the  pin  thus  fixed,  and  the  remainder  of  the  circle, 
will  serve  as  bases  for  continuing  the  evolution  of  the  wire, 
and  the  description  of  the  tooth.  The  same  position  of  the 
wire  will  show  the  outline  of  a  basis  proper  for  describing 
by  means  of  a  circle  rolled  on  it,  the  curve  which  must  be 
substituted  for  the  form  of  any  epicycloidal  tooth,  which 
might  have  been  described  by  causing  the  same  circle  to 
roll  on  the  simple  circumference  of  the  wheel  as  a  basis ; 
the  curved  part  of  the  tooth  beginning,  in  this  case,  at  the 
point  of  contact  first  mentioned. 

"  If  it  be  objected,  that  in  such  an  arrangement,  the 
equability  of  the  motion  would  be  lost,  and  a  shake  would 
be  created ;  it  may  be  answered,  that  the  inequality  would 
be  utterly  imperceptible  in  practice.  But  I  do  not  know, 
that  the  form,  thus  determined,  would  have  any  material 
advantage  over  teeth  made  as  short  as  possible,  or  so  cut 
away  as  not  to  act  before  the  passage  of  the  line  of  centres, 
which  may  easily  be  done  in  all  cases,  nearly  in  the  same 
way  as  you  have  shown  with  respect  to  epicycloidal  teeth* 

"  The  advantage  of  dividing  the  pressure  among  several 
teeth  ought  not  to  be  purchased  at  the  expense  of  an  in- 
crease of  friction,  since  the  property  of  greater  durability 


n 


CHAP.  Ul.]  ON    THE    TEETH    OF    WHEELS.  7 1 

may  be  obtained,  in  an  equal  degree,  by  simply  making  the 
wheels  thicker,  without  materially  adding  to  the  friction : 
uuJ  in  fact,  although  the  momentary  pressure  on  each 
tooth  may  be  leadened  by  dividing  it,  yet  its  duration  is  in- 
creased in  the  same  proportion. 

71 .  "I  must  beg  leave  to  observe,  that  the  form  proper 
for  the  teeth  of  a  pinion,  acting  on  a  rack,  is  the  involute 
of  a  circle,  and  not  a  cycloid.  The  cycloid  would  be  a 
proper  form  for  the  teeth  of  the  rack,  if  they  were  intended 
to  impel  the  pinion. 

"  It  has  been  remarked  that  the  form  of  the  involute  of 
a  circle  is  not  immediately  deducible  from  the  general 
principle  of  La  Hire  ■,  and  the  remark  is  strictly  true,  since 
the  curves,  formed,  according  to  that  principle,  fi-om  two 
contiguous  circles  as  bases,  could  not  act  on  each  other 
without  a  further  separation  of  the  centres,  which  would 
render  the  demonstration  inadequate.  But  I  have  ob- 
sen-ed  in  the  Additions  to  my  second  volume,  p.  x.  the 
principle  may  be  extended  to  any  other  curves,  as  well . 
as  circles  and  straight  lines  :  and  if  we  employ  an  equian- 
gular spiral,  instead  of  a  straight  line,  we  shall  have  the 
involutee,  exactly  as  they  are  recommended  for  practice." 


SUPPLEMENTARY  DEFINITIONS. 


72.  An  angle  is  the  inclination  of  two  Unes  to  one  an- 
other which  meeting  do  not  lie  in  one  line. 


73.  A  triangle  is  a  figure  contained  by  three  straight 


72 


ON   THE   TEETH   OF  WHEELS.  |^CHAP.  lU. 


74.  A  circle  is  a  plane  figure  contained  by  one  line, 
whicli  is  called  the  circumference,  and  is  such,  that  all 
straight  lines  drawn  from  a  certain  point  within  the  figure 
to  the  circumference,  are  equal  to  one  another,  and  this 
point  is  called  the  centre  of  the  circle. 


75.  The  radius  of  a  circle,  is  a  straight  line  drawn  from 
the  centre  to  the  circumference.  The  word  radii  is  used, 
when  more  than  one  such  line  is  spoken  of. 


76.  The  diameter  of  a  circle,  is  a  straight  line  drawn 
through  the  centre,  and  terminated  both  ways  by  the  cir- 
cumference. 


77-  The  arc  of  a  circle  is  any  part  of  its  circumference. 


GBAF«  nu}  ON  THE  TEETH  OF  WHEELS.  73 

78.  A  chord  of  an  arc,  is  a  straight  Ime  joming  the  two 
extremities  of  the  arc 


79«  A  tangent  of  a  circle  is  a  straight  line,  which  pasf 
through  a  point  in  the  circumference  without  cutting  it 


80.  A  polygon  is  a  figure,  having  more  than  four  sides. 
The  term  is  seldom  applied  to  figures  that  have  less  than 
fiye  sides. 


81.  Parallel  straight  lines  are  such  as  are  in  the  same 
plane,  and  which,  heing  continued  ever  so  far  either  way, 
never  meet 


83.  The  word  perpendicular  is  the  same  with  square,  as 
used  by  workmen. 

In  order  to  draw  from  a  given  point,  a,  in  a  given  line 
B  c,  another  line  perpendicular  to  it. 

Take  a  e,  equal  to  a  f,  and  from  the  points  f  and  £,  with 
any  radius  greater  than  a  b,  make  the  intersection  d  ; 
draw  D  A,  which  will  be  perpendicular  to  b  c. 


74 


OK  THE  TEETH  OF  WHEELS*  [cHAP.  HI. 


y<ru 


\ 

p  ~  E 

83.  If  it  be  required,  from  the  end  of  a  given  straight 
line  A  B,  to  raise  a  perpendicular, 

Take  the  point  c  nearer  to  a  than  b  ;  about  the  centre 
c,  with  the  radius  c  a,  describe  the  circle,  e  a  d  ;  through 
the  points  e  and  c,  draw  the  Une  £  d,  and  join  a  d,  which 
will  be  perpendicular  to  A  B ;  and  bad,  is  called  a  right  angle. 


E  V 


7^  A 


84.  To  let  fall  from  a  given  point  a,  a  perpendicular 
upon  a  given  straight  line  b  c. 

About  the  given  point,  describe  a  circle  cutting  b  c  in  e 
and  F ;  from  the  points  e  and  f,  make  the  secticm  n,  an4 
draw  the  line  a  d  from  a  towards  n,  and  a  d  is  the  pw- 
pendicular  required. 


B- 


^ 


7f 


MAP.  III.]]  OV    THE   TEETH    OF   WHEELS.  7^ 

85.  A  cylinder  is  a  body  having  two  flat  surfaces,  and 
one  circular.     For  instance,  a  roller  is  a  cylinder. 


cone  is  a  solid  body,  of  which  the  base  is  a  circle, 


and  which  ends  in  a  point. 


m 


K 

^V  87-  ^'^elocity  is  a  term  equivalent  to  speed. 

88.  Mr.  Smeaton  thus  defines  the  term  power :  "  The 
word  power,  as  used  in  practical  mechanics,  I  apprehend 
to  signify  the  exertion  of  strength,  gravitation,  impulse,  or 
pressure,  compounded  with  motion,  to  be  capable  of  pro- 
ducing an  eflect ;  and  that  no  eflect  is  properly  mechanical, 
but  what  requires  such  a  kind  of  power  to  produce  it."* 

89-  A  proposition  is  a  sentence  in  which  any  thing  is 
affirmed. 

90.  A  corollary  is  an  inference  or  deduction. 

91.  The  extract  from  Smeaton's  works  in  Art.  88,  con- 
veys very  little  information  respecting  power ;  and  yet  it  is 
necessary  that  everj-  mechanic  should  have  correct  ideas 
on  this  subject,  which  will  be  a  sufficient  reason  for  intro- 
dudug  a  further  explanation  here. 


iti>B'5  Minecllancous  Papers,  p.  30.     See  An.  91. 


76  ON   THE   TEETH   OF  WHEELS.  [CHAP.  HI. 

92*  Power  is  the  general  term  for  that  which  causes 
motion  or  rest  For  hodies  in  nature  are  in  a  state  of  rest, 
only,  when  the  opposing  powers  acting  upon  them  are  in 
equilibrium. 

But  this  general  term,  power,  is  divided  into  several 
particular  ones  according  to  the  circumstances  under  which 
it  acts. 

93.  When  power  is,  or  can  be,  balanced  at  rest,  it  seems 
to  be  most  proper  to  call  it  force ;  but,  to  distinguish  more 
precisely  the  circumstances  of  its  action,  it  is  necessary  to 
employ  the  simple  terms,  weighty  pressure^  and  stress,  and 
also  the  compound  terms,  force  of  attraction,  force  of  gra- 
vity, cohesive  force,  centripetal  force,  centrifugal  force,  and 
others  of  a  like  nature. 

94.  But  when  a  body  is  in  motion,  its  power,  at  any  in- 
stant, or  at  any  point  in  its  path,  is  usually  termed  fna- 
mentum,  or  moving  fprce,  or  quantity  of  motion.  It  is  this 
species  of  power  which  Sir  Isaac  Newton  makes  the  object 
of  his  second  definition.  (Mathematical  Principles  of  Natu- 
ral Philosophy,  Book  I.)  Some  writers  propose  to  use  the 
term  energy  instead  of  momentum,  (See  Edin.  Rev.  voL 
xii.  p.  ISO,)  but  there  does  not  appear  to  be  sufficient  rea- 
son for  adopting  it,  the  other  having  been  in  a  consider- 
able degree  restricted  to  this  species  of  power*. 

Here  we  take  the  liberty  of  remarking,  that  neither 
the  measure  of  momentum  nor  that  of  any  other  kind  of 
power,  has  any  relation  whatever  to  time ;  for  momentum 
simply  expresses  the  quantity  of  power  in  a  moving  body  at 
a  particular  instant,  without  reference  to  the  rate  of  accu- 
mulation, or  to  the  efiect  it  would  produce ;  and  similar 
remarks  apply  to  other  species  of  power. 

95.  It  is  further  necessary,  both  for  practical  and  scien- 

*  The  tenn  energy  has  also  been  applied  to  the  product  of  the  mass  of 
the  body  into  the  square  of  its  velocity.  See  Dr.  Young's  Nat.  Phil.  VoL 
II.  Art.  347. 


HAP.    HI.3  ON    THE    TEETH    OF    WHEELS.  77 

tlBc  purposes,  to  have  a  term  to  designate  that  power  which 
is  equivalent  to  momentum,  wlien  the  velocity  is  uniform. 
Smealon  employed  the  term  rnevlianical  power  for  this  piu-- 
po9e  ;  and  since  this  term  is  sanctioned  by  the  language  of 
all  writers  on  the  first  principles  of  meclianics ;  and  the 
simple  machines,  by  means  of  which  such  power  is  modified 
to  produce  the  desired  effect,  have  always  been  called  the 
mechanical  powers*.     We  think  it  will  be  found  desirable 

use  the  term  mechanical  power  in  preference  to  any 
«ther  that  has  been  proposed.  The  term  impetus  is  ob- 
jectionable, because  it  indicates  a  degree  of  violence  in  the 
action  of  power,  which  does  not  agree  with  what  takes 
place  in  the  most  common  applications  of  mechanical  powei-. 
And  it  is  questionable,  whether  its  proposer  did  not  intend 
it  to  be  a  measure  of  effect. 

We  must  now  attempt  to  inform  the  reader,  more  par- 
ticularly, of  the  circumstances  to  which  these  different  mo- 
difications of  power  apply,  and  in  so  doing,  we  shall  have 
occasion  to  place  a  most  interesting  department  of  mechani- 
cal science  in  a  different  light  from  what  it  has  been  regarded 
by  my  predecessors. 

JK>.  Force  is  immediately  comparable  with  the  weight  of 
a  quiescent  body.  Its  intensity,  direction,  and  equilibrium, 
are  the  proper  objects  of  that  part  of  mechanics  called  sta- 
tics, or  hydrostatics,  and  aerostatics,  when  the  body  exert- 
ing force  is  fluid. 

97-  Alomentuin,  or  the  force  of  a  moving  body,  is  pro- 
portional to  the  quantity  of  matter  in  the  body,  multiplied 
by  its  velocit)'  at  that  instant  when  the  comparison  is  made. 
Its  rate  of  increase  and  decrease,  its  direction,  and  equili- 
briara,  are  the  proper  objects  of  those  parts  of  mechanics 
called  dvnamics  and  bydrodjiiamics.      In  fact,  statics  is 

*  See  Dr.  Jamieson's  Mecbanics  for  Pmctical  Men,  oomprisiDg  Treatises 
00  the  CoiD[Hieition  and  Hesolution  of  ForceE,  the  Centre  of  Gravity,  and 
the  Mcchiuiic»I  Powen. 


78  ON   THE   TEETH   OF   WHEELS.         [CHAP.   III. 

only  that  particular  case  of  dynamics  when  the  velocity  is 
nothing.  In  like  manner  we  simplify  an  important  part  of 
the  science  of  mechanics  hy  separating  all  prohlems  in  which 
the  velocity  is  miiform ;  hecause  in  that  case  the  length  of 
the  line  the  body  moves  over,  is  proportional  to  the  velo- 
city. 

98.  Mechanical  power  then  is,  a  particular  name  for  the 
momentum  of  a  body  in  uniform  motion  ;  in  that  case,  it  is 
proportional,  to  the  quantity  of  matter  in  motion,  multiplied 
by  the  length  of  the  line  through  which  it  acts ;  conse- 
quently,  in  aU  problenis  where  the  motions  are  uniform, 
(and  there  can  be  no  difficulty  in  distinguishing  such  pro- 
blems,) this  measure  of  power  may  be  employed,  and  its 
motion,  equilibrium,  and  direction,  determined  accordingly. 
Every  person  conversant  with  the  management  of  such  pro- 
blems must  be  aware  of  the  advantage  of  this  mode  of  in- 
vestigation ;  it  applies  to  the  motion  of  water-wheels,  of 
wind-miUs,  of  rivers,  the  resistance  of  fluids,  &c.  &c,  and 
in  general  to  the  motion  of  machines.  It  has  often  been 
partially  applied  in  considering  the  equilibrium  of  mechani- 
cal powers,  (see  Wood^s  Mechanics,  prop.  xxx.  and  xxxi.,) 
but  we  are  not  aware  of  its  having  been  previously  pointed 
out  as  a  general  principle,  with  the  object  of  forming  a  dis- 
tinct branch  of  mechanics.  Some  writers  have  confounded 
measure  of  power  with  measure  of  effect  so  far  as  to  sup* 
pose,  that  mechanical  power  is  identical  with  the  quantity 
of  matter  multiplied  into  the  square  of  its  velocity;  we 
hope  the  true  nature  of  mechanical  power  is  here  so  de« 
fined  as  to  prevent  a  recurrence  of  a  like  mistake. 

It  is  much  to  be  regretted  that  power  has  not  been  made 
the  basis  of  all  mechanical  science,  in  the  place  of  motion; 
for  motion  is  merely  an  affection  or  mode  of  matter  acted 
upon  by  unbalanced  force.  For,  in  the  practical  applicsr 
tion  of  mechanics,  it  would  prevent  error ;  and  in  the  theory 
of  mechanics,  that  interesting  phenomenon»  the  aocumuhh 


CHAP.   III.]  ON   THE   TEETH    OF   WHEELS.  79 

Hon  of  power ^  must  have  been  forced  upon  the  attention  of 
philosophers. 

99*  Buchanan  had  long  employed  himself  in  making 
a  collection  of  facts  respecting  wheels  actually  in  use  in 
millwork,  and,  by  arranging  them  agreeably  to  his  own 
views,  draw  such  useful  practical  inferences  as  might  bene- 
fit workmen  generally.  All  the  facts  he  had  been  able  to 
collect  and  arrange  will  be  found  in  the  following  Chapter. 

Time  and  other  circumstances  did  not  allow  him  to  enter 
more  minutely  into  the  subject.  Yet  these  hints,  even  in 
their  present  state,  may  lead  to  a  fuller  investigation,  and 
they  will  not  be  altogether  without  some  advantage,  espe- 
cially  as  nothing  exactly  of  the  same  kind  has  hitherto  been 
published  in  this  country. 

With  respect  to  the  elementary  propositions  which  guide 
U8  in  this  inquiry  into  the  proportional  strength  of  the 
teeth  of  wheels,  Buchanan  did  not  enter  into  their  demon- 
strations. To  the  artisan,  unacquainted  with  mathematics, 
they  would  be  unintelligible ;  and  the  mathematician  can 
either  demonstrate  them  himself,  or  have  recourse  to  those 
elementary  writings  where  the  demonstrations  may  be  found : 
of  these  last,  as  being  more  generally  accessible,  we  refer 
to  "  Emerson's  Mechanics,"  quarto  edition ;  and  to  the 
volumes  of  the  "Encyclopaedia  Metropolitana"  comprising 
machinery,  and  edited  by  Professor  Barlow,  of  Woolwich ; 
also  Dr.  Robinson's  "  Mechanical  Philosophy,"  as  edited 
by  Brewster,  and  the  excellent  paper  of  Mr.  Willis,  on  the 
**  Teeth  of  Wheels,"  published  originally  in  the  second 
volume  of  the  Transactions  of  the  Institution  of  Civil  En- 
gineers, London,  1838,  and  added  as  an  Appendix  to  this 
work  of  Robertson  Buchanan. 


CHAPTER  IV. 

A  PRACTICAL  INQUIRY  RESPECTING  THE  STRENGTH  AND 
DURABILITY  OF  THE  TEETH  OF  WHEELS  USED  IN  MILL- 
WORK. 

100.  Having  treated  of  the  forms  of  the  teeth  of  wheels, 
we  come  now  to  consider  their  proportional  strength  with 
relation  to  the  resistance  they  have  to  overcome. 

We  are  aware  that,  owing  to  a  great  variety  of  circmn- 
stances,  this  subject  is  involved  in  much  difficulty,  and  that 
it  is  no  easy  task  to  form  any  general  rule  with  regard  to 
the  pitches  and  breadths  of  the  teeth  of  wheels.  We  do 
not  pretend  to  more  than  a  mere  approximation  towards 
general  rules ;  yet,  were  this  judiciously  done,  we  are  of 
opinion,  that  it  might  be  useful  to  the  millwright,  who  has 
not  had  leisure  or  opportunity  for  scientific  inquiries.  A 
rule,  though  not  absolutely  perfect,  is  better  in  all  cases, 
than  to  have  no  guide  whatever. 

And  it  is  too  evident  to  require  proof,  that  it  is  essential 
to  the  beauty  and  utility  of  any  machine,  that  the  strength 
and  bulk  of  its  several  parts  be  duly  proportioned  to  the 
stress  or  wear  to  which  the  parts  may  be  subject. 

Some  general  observations  on  the  wheel  work  of  mills, 
will  serve  greatly  to  simplify  our  inquiries  on  the  subject. 


GENERAL   OBSERVATIONS  ON   THE  WHEEL  WORK  OP  MILLS. 

101.  Mistaken  attempts  at  economy  have  often  prompted 
the  use  of  wheels  of  too  small  diameter.  This  is  an  evil 
which  ought  carefully  to  be  avoided.     Knowing  the  pres- 


CUAP.  IV.]  ON    THE    TEETH    OF    WHEELS. 

stire  on  the  teeth,   we   cannot  with  propriety  reduce  the 
(iiameter  of  a  wheel  below  a  certain  measure. 
Suppose,  for  instance,  a  water  wheel  of  20  horses'  power, 
I      moving  at  the  pitch  line  with  a  velocity  of  3^  feet  per  second. 
'     It  IB  known,  that  a  pinion  of  4  feet  diameter,  might  work 
uito  it,  without  impropriety ;   hut  we  also  know,   that  it 
Hould  be  exceedingly  improper  to  substitute  a  pinion  of 
only  one  foot  diameter,  although  the  pressure  and  velocity 
at  the  pitch  lines  in  both  cases  would  he,  in-  a  certain  sense, 
the  same.     In  the  case  of  the  small  pinion,  however,  a 
much  greater  stress  would  be  thrown  on  the  journeys  (or 
journals)  of  the  shaft.     Not,  indeed,  on  account  of  torsion 
or  twist,  but  on  account  of  transverse  strain,  arising,  as 
well  from  greater  direct  pressure,  as  from  the  tendency 
which  the  oblique  action  of  the  teeth,  particularly  when 
somewhat  worn,  would  have  to  produce  great  friction,  and 
to  force  the  pinion  from  the  wheel,  and  make  it  hear  harder 
on  the  journals.     The  small  pinion  is  also  evidently  liable 
to  wear  much  faster,  on  account  of  the  more  frequent  re- 
currence of  the  friction  of  each  particular  tooth. 

That  these  observations  are  not  without  foundation,  is 
knairn  to  millwrights  of  experience.  They  have  found  a 
great  saving  of  power,  by  altering  corn  mills,  for  example, 
from  the  old  plan  of  using  only  one  wheel  and  pinion,  (or 
trundie,)  to  the  method  of  bringing  up  the  motion,  by 
means  of  more  wheels  and  pinions,  and  of  larger  diameters 
and  finer  pitches. 

The  increase  of  power  has  often  by  these  means  been 
nearlv  doubled,  while  the  tear  and  wear  has  been  much 
lussened ;  although  it  is  evident,  the  machinery,  thus  al- 
tered, was  more  complex. 

The  due  consideration  of  the  proper  communication  of 
ihe  original  power,  is  of  great  importance  for  the  construc- 
Uoo  of  mills  on  the  best  principles.       It  may  easily   be 


82  ON   THE   TEETH    OF   WHEELS.  f  CHAP.  IV. 

seen,  that  in  many  cases,  a  very  great  portion  of  the  ori- 
ginal power  is  expended,  before  any  force  is  actually  ap- 
plied to  the  work  intended  to  be  performed. 

Notwithstanding  the  modem  improvements  in  this  de- 
partment, there  is  still  much  to  be  done.  In  the  usual 
modes  of  constructing  mills,  due  attention  is  seldom  given 
to  scientific  principles.  It  is  certain,  however,  that  were 
these  principles  better  attended  to,  much  power,  that  is 
unnecessarily  expended,  would  be  saved.  In  general,  this 
might  be  in  a  great  measure  obtained,  by  bringing  on  the 
desired  motions  in  a  gradual  manner,  beginning  with  the 
first  very  slow,  and  gradually  bringing  up  the  desired  mo- 
tions, by  wheels  and  pinions  of  larger  diameters.  This  is 
a  subject  which  should  be  well  considered  before  we  can  de- 
termine, in  any  particular  case,  what  ought  to  be  the  pitch 
of  the  wheels.  In  the  case  above  alluded  to,  where  the 
supposition  is  a  pinion  of  4  feet  diameter,  or  of  1  foot  dia- 
meter ;  it  is  obvious,  that  the  same  pitch  for  both  would 
not  be  prudent.  That  for  the  small  pinion,  ought  to  be 
much  less  than  that  which  might  be  allowed  in  the  case  of 
the  larger  pinion.  It  is  also  equally  ob\dous,  that  the 
breadth  of  the  teeth,  in  the  case  of  the  small  pinion,  ought 
to  be  much  greater  than  that  in  the  case  of  the  larger 
pinion. 

102.  It  is  evident,  however,  that  although  great  ad- 
vantage may  often  be  derived  from  a  fine  pitch,  that  there 
is  a  limit  in  this  respect,  as  also  with  regard  to  the  breadth. 
We  shall  endeavour  to  find  some  trace  of  this  limit  in  what 
follows ;  and  that  we  may  the  better  do  this,  we  shall  call  in 
the  aid  of  propositions,  which  are  true  with  respect  to  pieces 
of  timber,  or  metal,  subjected  to  ordinary  cases  of  pressure. 
It  is  allowed,  that  they  cannot  here,  in  strictness,  be  de- 
monstratedj  as  applicable  to  wheelwork.  Yet  they  wiU,  for 
want  of  better  light,  serve  at  least  to  prevent  any  material 


niAP.   rv.]  ON    THE   TEETH    OF    WHEELS.  S.S 

[)ractical  error  with  regard  to  the  strength  of  the  teeth  of 
ffhech.     For  it  is  to  be  remembered,  that  we  are  not  so 

I  much  here  in  search  of  truths  of  curious  or  profound  mathe- 
matical speculation,  as  of  that  kind  of  evidence  of  which 
the  subject  admits,  and  which  may  l)e  sufficiently  satisfac- 
tory for  any  practical  purpose. 

It  most,  however,  be  understood,  that  we  suppose  the 
diameters  made  sufficiently  great  to  prevent  the  evils  which 
lie  have  already  noticed,  and  in  the  annexed  table,  (Art. 
120,}  are  some  examples  in  actual  use,  which  have  been 
foimd  in  practice  sufficiently  durable.  We  would  particu- 
larly  recommend  attention  to  those  of  Boidton  and  Watt, 
whose  most  extensive  practice,  as  well  as  scientific  know- 
ledge,  renders  their  work  a  model  well  worthy  the  atten- 
tion of  millwrights. 

As  cast-iron  pinions  are  now  generally  used,  and  as  the 
teeth  of  the  pinion  are  most  subject  to  wear,  I  think  we 
are  safe,  in  the  present  inquiry,  in  considering  them  all  as 
cast-iron. 

TTie  laws  to  which  we  have  alluded  in  this  investigation 
are  these : — 

I 

^^B  lOS.  The  strength  of  any  piece  of  timber,  or  metal, 
^^g^Qse  section  is  a  rectangle,  is  in  direct  proportion  to  the 
'     hraadth,  and  as  the  square  of  the  depth. 

Let  B  D  be  any  beam,  placed  horizontally,  and  fixed  at 
ihc  cod  Bc,  and  let  afg  be  the  perpendicular  section  in 
which  the  fracture  is  supposed  to  take  place.  Divide  the 
depth  A  P  into  an  infinite  number  of  equal  parts  at  n,  b,  c,  rf, 
,  whose  aj^^egate  is  n  =af,  and  through  each  of  those 

G  Q 


PRINCIPLES  OF  PROPORTIONING  THE  STRENGTH  OF 
TEETH  OF  WHEELS. 


PROPOSITION    I 


S4> 


ON   THE   TEETH   OP  WHEELS.  [^CHAP.  IT. 


divisioDS  suppose  sbraight  lines  to  be  drawn  parallel  to  fg 
the  upper  side  of  the  beam.  Then  let  any  force  be  applied 
at  p  in  the  direction  dp  to  break  the  beam  at  af.    Now, 


since  the  strength  of  the  timber  is  nothing  but  the  force  by 
which  the  particles  cohere  together,  the  breaking  of  the 
timber  is  nothing  but  overcoming  this  force  and  separating 
the  parts  from  one  another. 

Let  the  force  of  cohesion  of  any  one  of  the  parts  be  de- 
noted by  unity,  and  imagine  dAO,  qa6,  qac,  &c.,  to  be  so 
many  bent  levers  whose  fulcrum  is  at  a  ;  we  have  then 
to  inquire  what  will  be  the  sum  of  all  the  forces  applied  at 
Q  the  extremity  of  the  levers,  to  break  the  beam  at  a. 

Now  by  the  property  of  the  lever,  the  power  applied  at  q 
to  equal  or  overcome  the  resistances  at  a,  a,  b,  c,  &c.,  will  be 

0      Aa      \b     AC     o         iAF  r  ,1  T.'i- 

— ,  — ,  — ,  — ,  &c.,  to  —  J  or  because  the  cohesive  force 

AQ    AQ     AQ    AQ  AQ 

of  any  one  of  the  filaments  in  the  section  is  represented  by 

unity,  it  wUl  be 

0      1      2      3     0       .     n 
— ,  — ,  — ,  — ,  &c.,  to — * 

AQ    AQ     AQ    AQ  AQ 

Consequently,  the  effect  of  all  the  forces  applied  at  q  to 
break  the  beam ;  that  is,  the  whole  strength  of  the  beam 
will  be,  as 


-L  (0  +  1  +  fi 

AQ 


J  +  . . . .  +  ra),  or  as 

AQ 

Therefore,  since  aq  is  given,  the  strength  of  the  beam  is 


IV.] 


ON    THE   TEETH    OF    WHEELS. 


as  the  square  of  the  depth,  or  as  af*  =  n*.  Now,  it  is  ob- 
vious, that  if  the  breadth  fg  be  increased  iii  any  proportion, 
the  strength  of  the  parts  must  be  increased  in  the  same 
proportion.  So  that  the  absolute  lateral  strength  of  the 
beam,  will  be 
^L  as  FG  X  af'. 

104.  Hence  may  be  inferred,  that  the  strength  of  teeth 
of  wheels,  moving  at  the  same  velocity,  and  under  the  same 
circumstances,  is  directly  in  proportion  to  their  breadth, 
and  as  the  square  of  their  thickness.  Thus,  for  example, 
if  we  double  the  breadth,  we  only  double  the  strength ; 
but  if  we  double  the  thickness,  in  other  words,  double  the 
pitch,  keeping  the  original  breadth,  we  increase  the  strength 
four  times. 

For  although  when  wheels  are  working  accurately,  the 
stnun  is  at  the  same  time  divided  over  several  teeth,  yet  as 
a  very  small  inaccuracy,  or  even  the  interposition  of  any 
small  body,  such  as  a  chip  of  wood  or  stone,  throws  the 
whole  stress  upon  a  single  tooth  in  practice  ;  therefore,  and 
in  order  to  simplify  this  case,  we  may  consider  the  strength 
of  a  single  tooth,  as  resisting  the  pressure  of  the  whole 
work. 

But  as  the  length  of  teeth  commonly  varies  with  the  pitch, 
this  circumstance  must  be  taken  into  account,  and  the  most 
ample  view  we  can  take  of  it  seems  to  be,  that  of  having 
the  strain  of  each  tooth,  thrown  all  to  the  outward  extre- 
mitv  ;  we  have  then  the  following  proposition  to  guide  this 
part  of  our  inquiry. 


105.  Ifanifj'vrce  be  applied  laierallj/  to  a  lever  or  beam, 
e  stress  upon  any  place,  is  directly  as  the  Jbrce  and  its 
iejrom  that  place. 


86  ON   THE   TEETH    OF   WHEELS.  [CHAP.  IV. 

For   suppose  paf   to  be   a   bent  lever;   it  is  evident 
that  the  greater  the  power  applied  at  p,  the  greater  is  the 


force  exerted  at  f  to  separate  the  particles  of  the  beam  in 
that  place.  Also  the  greater  the  distance  a  p,  the  greater 
effect  has  any  power  applied  at  p  to  overcome  the  cohesion 
of  the  wood  at  f  ;  and  therefore,  the  whole  stress  depends 
upon  both. 


PROPOSITION    III. 


106.  The  pitch  being  the  same^  the  stress  is  inversebf 
as  the  velocity. 

This  is  obvious,  for  the  teeth  of  wheels,  and  the  wheels 
themselves,  which  act  with  greater  force,  must  be  propor- 
tionally stronger ;  and  in  any  combination  of  wheels  and 
axles,  the  strength  must  diminish  gradually  from  the  weight 
to  the  power,  so  that  at  every  part  it  may  be  reciprocally 
as  the  velocity  of  that  part. 

For  example — ^if  the  pitch  lines  of  one  pair  of  wheels  be 
moving  at  the  rate  of  6  feet  in  a  second,  and  another 
pair  of  wheels,  in  every  other  respect  under  the  same  cir- 
cumstances,  be  moving  at  the  rate  of  3  feet  in  a  second,  the 
stress  on  the  latter  is  double  of  that  on  the  former. 

107.  This  proposition  is  true  only  in  the  wheels  of  the 
same  machine.  To  render  it  universal,  the  first  movers  of 
all  machines  must  be  reduced  to  the  same  standard ;  which 
may  be  done  as  follows,  where  the  horse  power  is  supposed 
to  be  the  standard. 

If  p  be  the  power  of  the  first  mover,  in  anv  machine. 


CHIP.   IV.]         ON    THE    TEETH    OF    WHEELS.  8? 

and  V  lis  velocity ;  also,  let  w  be  the  velocity  of  any  part 
(HI  which  it  is  necessary  to  determine  the  stress.     Then, 

as  ('  :  v::  p  :  stress  =  - — .    (Wood's    Mechanics,     Prop. 

I' 
XWl.) 

\W  taking  the  same  value  of  the  horse  power  as  is 
used  throughout  this  Work;  that  is  200 lbs.  moved 
at  the  rate  of  3|    feet  per  second,  and  using  h    to  re- 


present the  number  of  horses,  we  have 


2tX)H  X  3| 


Bv  using  i- =  the  stress,  we  get  rid  of  the  fractions, 

nd  have  a  sufficiently  accurate  measure  of  the  force. 
Hence,  if  the  power  of  a  machine  be  equal  to  any  number 
k  of  horses,  the  stress  at  any  pitch  line  of  which  the  velo- 

j  is  V  feet  per  second  will  be  =  i— 

[' 

In  a  hke  manner,  the  stress  at  the  surface  of  any  journal 
r  shaft  may  be  found. 
But,  in  those  cases  where  the  same  first  mover  gives 
lotion  to  different  trains  of  machinery,  the  stress,  at  any 
part  of  any  one  of  them,  should  be  measured  by  the  greatest 
uumber  of  horses'  power  necessarj'  to  perform  the  work  as- 
Tied   to  that  train.      That  is,  if  ii  be  the  number  of 

i  that  could  perform  the  work,  then  - — —  =  the  stress 

V 

\  anv  point  in  the  train  moving  with  the  velocity  r. 
'  We  shall  confine  our  attention  for  the  present  to  wheels 
iBving  cast-iron  teeth  ;  and  in  order  to  take  experience  as 
r  guide,  several  examples  in  the  annexed  tables,  actually 
i  use,  are  selected. 

I  The  pitch,  velocity,  and  strain,  are  all  stated  ;  the  strain 
I  measured  bv  the  horsed  power,  at  which  the  resistance 
\  valued.     Horses'  power  is  a  term  now  in  general  use,  lo 


88  ON   THE   TEETH   OF   WHEELS.  [CHAP.  IV. 

express  the  force  required,  in  order  to  drive  any  kind  of 
mill,  and  it  may  be  proper  here  to  give  some  further  ac- 
count of  it 


horses'  power. 


108.  Although  horses  are  not  all  of  one  strength,  yet 
there  is  a  certain  force  now  generally  agreed  upon  among 
those  who  construct  steam  engines,  which  force  is  denomi- 
nated a  horses  power ^  and  hence,  steam  engines  are  dis- 
tinguished, in  size,  by  the  niunber  of  horses*  power  to  which 
they  are  said  to  be  equaL 

The  measure  of  a  mechanical  effect  equal  to  a  horse's 
power,  has  been  much  disputed:  this  I  believe  to  be  a 
matter  of  little  consequence,  if  the  measure  be  generally 
understood,  since  there  is  no  such  thing  as  bringing  this 
into  any  real  measure.  Some  horses  will  work  double  of 
others,  and  horses  of  one  country  will  work  more  than  those 
of  another.  Desagulier's  measure  is,  that  a  horse  will 
walk  at  the  rate  of  9\  miles  per  hour,  against  a  resistance 
of  200  lbs.  *  and  which  gives,  as  a  number  for  comparisons, 
44,000 ;  that  is,  the  raising  of  1  lb.  44,000  feet  in  a  minute; 
or,  what  amounts  to  the  same,  the  raising  of  44,000  lbs.  1 
foot  in  a  minute. 

Emerson's  measure  is  the  same  as  Desaguliers's,  (see 
Emerson's  Mechanics,  p.  178,)  and  Mr.  Smeaton's  result 
is  22916  lbs.  under  the  same  circumstances!. 

James  Watt  found,  from  repeated  experiments,  that 
33,000  lbs.  1  foot  per  minute,  was  the  average  value  of  a 
horse's  power;  but  his  engines  were  calculated  to  work 
equal  to  44,000  lbs.  1  foot  per  minute. 

*  When  working  8  hours  a  day,  (Desaguliere's  Course  of  Experimental 
Philosophy,  Vol.  I.  p.  241,)  2^  miles  per  hour  is  equal  to  220  feet  per 
minute,  or  3J  feet  per  second. 

t  Reports,  Vol.  I.  p.  229.  But  Desaguliers  gives  the  immediate  power 
of  a  horse,  Smeaton  the  effect  of  that  power  applied  to  raise  water :  hence 
the  friction  of  the  machinery  should  he  added  to  Smeaton's  horse's  power. 


nt.ii'.  IV.3 


ON    THE    TEETH    OF    ■ 


89 


But,  that  he  allows  only  ^3,000  in  his  calculations,  ap- 
'      pliej  to  mills,  considering  the  difference  as  being  lost  in 
the  friction  of  the  engine  itself. 

109.  It  is  common  in  practice,  to  reckon,  that  it  requires 
one  horse*s  power  to  drive  100  spindles  with  preparation 
'if  cotton  water  twist. 

110.  One  thousand  spindles  with  preparation  cotton 
mule  yam. 

HI.  Sevcnty.fivc  spindles  with  preparation  flax  yam. 
We  beg  leave  here  to  make  the  following  extract,  on  the 
subject  of  animal  force,  from  Dr.  Young's  Natural  Philo- 
sophy, Vol,  II.  p.  1(j5. 

11 '2.  "In  order  to  compare  the  different  estimates  of 
llif  force  of  moving  powers,  it  will  be  convenient  to  take  a 
unit  which  may  be  considered  as  the  mean  effect  of  the 
labour  of  an  active  man,  working  to  the  greatest  possible 
advantage,  and  without  impediment  j  this  will  be  found, 
upon  a  moderate  estimation,  sufficient  to  raise  10  pounds 
in  feet  in  a  second,  for  10  hours  in  a  day ;  or  to  raise  100 
[>ounds,  which  is  the  weight  of  1'2  wine  gallons  of  water,  1 
ftMit  in  a  second,  or  36,000  feet  in  a  day,  or  3,000,000 
pounds,  or  4-32,00{.)  gallons,  1  foot  in  a  day ;  this  we  may 
call  a  force  of  1,  continued  3G,000." 

113.   Immediate  force  of  men  and  horses,  without  de- 
■^  duction  for  friction. 


"A  tnan  of  orJinary  strength  can  turn  a  winch, 

nnlb  a  force  of  30  pouiidB.  and  with  a  velocity  of 

3|  feet  in  l"  for  10  hours  a  day."— Deaaguliers 

"Two   men    working  at   a   windlass,   ivith 

handle?  at  right  angles,  can  raise  70  pounds  more 

c««ilr  than  one  can  raise  30."— Desaguliera 

"  For  a  "hort  time,  a  man  may  exert  a  force 
8f>  [KiundB,  with  a  fly,  when  the  mution  is 

!Uv  ijuieV." — Desaguliers 

*•  A  honw  can  draw,  with  a  forceof  200  pounds, 

,  ndlni  an  lioiir  for  6  lionrs  in  the  day 

M  Willi  s  force  of  2*0,  only  6  hours."— De- 


I 


90 


ON   THE   TEETH    OF   WHEELS.  [CHAP.   IV* 


114.  Performance  of  men  and  horses  by  machines. 


^'  A  man  can  raise,  by  a  good  common  pump, 
a  hogshead  of  water  10  feet  high  in  a  minute, 
for  a  whole  day." — Desaguliers 

^'  By  means  of  pumps,  a  horse  can  raise  250 
hogsheads  of  water  10  feet  high  in  an  hour." — 
Smeaton's  Reports* 

Force. 

Conti. 
nuance. 

Day*! 
Woii. 

•875 
'864, 

•  •  • 
8h. 

•875 

1 15.  The  power  of  men  and  horses  to  move  machines, 
has  very  frequently  been  made  a  subject  of  investigation  by 
writers  on  mechanics  ;  but  yet  it  appears  possible  to  con- 
sider it  in  a  different  manner,  which  will  furnish  results 
from  principles  somewhat  more  strictly  practical  than  those 
which  have  hitherto  been  made  the  basis  of  calculation. 

116.  It  is  almost  always  a  necessary  condition,  that  the 
moving  power  of  a  machine  be  sensibly  uniform,  and  con- 
sequently the  power  which  moves  it,  should  be  uniform,  or 
of  that  kind  termed  mechanical  power,  see  Art.  98. 

117-  Let  PD  be  the  mechanical  power  of  a  horse,  which 
can  be  continued  during  a  whole  day,  and  also  day  after 
day,  without  exhausting  its  strength.  Then,  since  the  ex- 
ertion  of  the  muscles  will  be  constant,  the  product  pd  will 
be  a  constant  quantity,  for  the  degree  of  exertion  will  not 
be  altered  by  altering  either  d,  the  distance  passed  over  in 
a  second,  or  the  force  p.  But,  let  d  be  the  distance  when 
the  force  p  is  the  least  possible  corresponding  to  the  man- 
ner of  action  the  machine  requires,  or  in  other  words,  when 
the  power  to  move  the  machine  to  produce  useful  effect 
would  be  nothing.  And  let  p  dhe  the  mechanical  power, 
when  the  distance  passed  over  in  a  second  is  d.  Then, 
PB^pdf  and,  rf(jo  — p)  =  the  mechanical  power  exerted 
on  the  machine,  which  is  to  be  the  greatest  possible.     But 

p  =  ^,  therefore  jo  (rf  —  — )  =  a  maximum. 

D  D 

»  Vol.  I.  p.  229. 


HAP-  t>'.3 


ON    THE   TEETH    OF   WHEELS. 


Now  it  may  be  shown  by  the  principles  of  maxima,  &c., 
(hat  this  quantity  is  a  maximum  when  rf  =  ^  d.      For 

(P 
putdng  the  expression  p  (d  —  —)  into  fluxions,  and  equat- 
ing the  fluxion  with  zero,  or  0,  we  get 


•idd 


0; 


I 

^und  consequently,  by  transposition  it  becomes 

^K  ^^  =  1,  or  d  =  k  D. 

^^H||HKfbre,  a  man  or  a  horse  acts  with  the  greatest  ad- 
^^Bp^  on  a  machine  when  he  moves  with  half  the  velocity 
■    Iw  rmild  continue  at,  were  the  effective  resistance  of  the 

machine  nothing. 
That  portion  of  the  mechanical  power  which  is  efficient 

in  impelling'  the  machine  will  be  ^  pd.    For  since  rf  =  -J  d, 

p=  —  =  ^/),  or2p=^;  hence  (f(^  —  p)  =^  D  (2p  — p) 

=   ^  PD. 

The  force  p,  and  the  distance  moved  through  d,  will 
each  vary  in  the  same  man  or  horse  according  to  the  man- 
ner of  spphing  the  force  ;  but  their  product  will  be  nearly 
a  constant  quantity.  In  any  case,  one  of  these  quantities 
may  be  determined  from  experience,  and  in  many  instances 
both  of  them,  and  that  one  may  always  be  supplied  by  cal- 
culation which  cannot  be  found  by  experience. 

118.  When  the  effective  force  is  nothing,  it  must  not  be 

leretooc]  that  a  roan  or  a  horse  is  acting  against  a  force 

lal  to  his  own  weight,  in  any  case  whatever;    for  as  has 

Wn  observed  by  Dr.  Young,  (Vol.  1.  p.  132,  and  212,)  in 

walking,  the  resistance  overcome  is  not  exactly  comparable 

weight,  and  the  same  may  be  remarked  on  other 

of  exerting  force. 

"Wben  a  man  ascends  vertically,  his  velocity  is  reduced  to 

>ut  one  half  of  his  horizontal  velocity,  indicating  that  he 


Hqnt 


92  ON   THE   TEETH   OF  WHEELS.        ^I^^^^*   ^* 

acts  against  a  double  resistance;  therefore  when  a  man 
ascending  a  ladder,  carries  a  load,  the  maximum  effect  will 
take  place  when  his  ascending  velocity  is  about  one  fourth 
of  the  velocity  he  can  walk  horizontally  without  a  load. 

A  man  of  ordinary  strength  will  not  be  able  to  walk, 
unloaded,  at  a  quicker  rate  than  3|  miles  an  hour,  if  this 
exertion  is  to  be  continued  for  10  hours  every  day.  In- 
deed, those  who  examine  the  subject  with  a  view  to  a  £edr 
average,  will  find  this  to  be  about  the  extreme  velocity  that 
can  be  continued,  without  injury,  for  any  considerable 
time. 

According,  therefore,  to  our  investigation,  a  man  ought 
to  move  with  half  this  velocity  to  produce  a  maximum 
effect ;  that  is,  at  the  rate  of  If  mile  an  hour,  which  is 
about  2^  feet  per  second. 

But  this  supposes  the  whole  load  to  be  the  useful  effect, 
whereas  part  of  it  must  consist  of  the  apparatus  employed 
to  carry  it,  or  the  friction  of  an  intermediate  machine,  or 
other  circumstances  of  a  like  nature.  About  one  fifth  of 
the  velocity  may  be  considered  equivalent,  at  an  average, 
to  the  force  lost  in  ftiction,  &c.,  in  all  cases;  in  many  it  will 
exceed  one  fifth.  Hence  the  maximum  of  useful  effect  will 
take  place  when  the  velocity  is  2  feet  per  second,  or  about 
11  furlongs  an  hour,  continued  for  10  hours  each  day. 

Smeaton  is  said  to  have  made  numerous  comparisons, 
from  which  he  concluded  that  the  mechanical  power  of  a 
man  is  equivalent  to  3750  lbs.  moving  at  the  velocity  of 
one  foot  per  minute  *  ;  and  taking  this  average  to  be  near 
the  true  one,  as  I  have  reason  to  conclude  it  is,  we  have 

."^'Ji^  31 -25  lbs.  Therefore,  we  make  the  average  me- 
2x60  ^ 

chanical  power  of  a  man  31*25  lbs.  moving  at  the  velocity 
of  2  feet  per  second,  when  the  useful  effect  is  the  greatest 
possible ;  or  half  a  cubic  foot  of  water  raised  two  feet  per 

♦  Art.  Water,  Rees's  Cyclopaedia. 


lAP.    IV.] 


ON     THE    TEETH    OF   WHEELS. 


93 


second ;  a  very  convenient  expression  for  hydrodjmamical 
inquiries. 

tif  a  man  ascend  a  vertical  ladder,  according  to  a  pre- 
ling  remark,  (p.  91,)  the  velocity  which  corresponds  to 
•  maximum  of  useful  effect  will  he  1  foot  per  second,  and 
the  load  double  that  which  he  carries  horizontally  ;  conse- 
quently the  average  of  useful  effect  is  6'2'5  lbs.  raised  one 
foot  per  second. 

Bricklayers'  labourers  in  London  ascend  ladders  with  a 
load  of  about  80  lbs.  besides  the  hod ;  sometimes  at  the 
rate  of  one  toot  per  second,  but  more  frequently  about  9 
inches  per  second. 

Ascending  stairs  is  more  fatiguing  to  the  muscles  of  the 
legs  than  ascending  a  ladder ;  and  therefore  the  useful 
effect  is  less,  till  a  person  has  become  accustomed  to  this 
kind  of  labour.  And  it  is  also  to  be  observed  that  the 
space  moved  over  is  increased,  unnecessarily,  except  where 
the  horizontal  distance  is  part  of  the  path  over  which  the 
load  is  to  be  moved. 

We  ought  not  to  be  surprised  at  the  opposite  conclusions 
here  obtained,  from  those  of  other  theoretical  inquirers, 
when  the  data  they  have  proceeded  from  are  considered. 
For  their  data  have  been  the  extremes  of  force  and  velo- 
cit}',  without  any  knowledge  of  the  true  laws  which  con- 
nect theni- 

1 19.  The  force  of  a  horse  is,  at  an  average,  about  equal 
to  tliat  of  six  men,  according  to  various  estimates ;  and  the 
rate  of  travelling  about  the  same,  perhaps  rather  less  than 
that  of  a  man,  when  his  exertion  is  continued  for  8  hours  j 
quently  the  velocity  corresponding  to  the  maximum 

set,  will  be  about  '2^  feet  per  second.  Whence,  the 
•  mechanical  power  of  a  horse  may  be  estimated  at 

7i  lbs.  moving  with  a  velocity  of  2^  feet  per  second,  or 

ubic  feet  of  water  raised  2^  feet  per  second.  The  day's 
L  being  8  hours. 


94  ON   THE   TEETH   OF   WHEELS.  [CHAP.  IV. 

This  estimate  of  the  power  of  a  horse  is  equal  to  28, 125  lbs. 
raised  one  foot  per  minute ;  nearly  a  mean  between  Watfs 
and  Smeaton's.  But  since  our  author  has  employed 
that  of  Desaguliers,  it  was  not  very  easy  to  make  a  change 
in  this  work ;  and  still  more  objectionable  to  employ  two 
measures  of  different  values. 

French  writers  use  as  a  dynamical  unit  a  given  measure 
of  water  raised  through  a  given  space ;  and  the  Americans 
use  a  similar  measure.  In  my  opinion,  it  is  preferable  to 
make  a  horse's  power  the  dynamical  unit ;  because,  a  prac- 
tical man  has  a  more  correct  idea  of  the  quantity  of  me- 
chanical power  expressed  by  this  unit,  than  he  can  have  of 
any  one  less  frequently  under  his  observation ;  besides,  it 
is  a  power  often  employed  to  move  machines ;  and  there- 
fore is  a  familiar  measure  of  comparison. 


•  3                  ON    THE    TEETH    OF    WHEELS.                                      9^            ^^H 

TABLB  OP   PITCHES  01'   WHEELS   IN    ACTUAL    USB   IN   MILLWOHK. 

■ 

=i 

s 

u. 

Wheel. 

Pinion. 

1 

1 

1 

^1 

i"i 

«-. 

, 

1 

I 

Is 
Is 

i 
f 

y 

y 

1 

•s 

1 

1 

1 

1 

1 

ll 

ill 

1 

li 

pi 

sis 

■ 



= 

£ 

p. 

r. 

a 

0 

£ 

^H 

fe^' 

10 

^ 

51 

"■.  1°' 

'\'.°' 

55 

■ 

feed',  B 

30 

3 

101 

W 

"9J24 

"3 

3is"47 

3'41 

3-as 

4-489 

riml.   C 

15 

3 

6 

■204 

4j 

16    3 

44 

20 

3  "e 

4- 

3-8 

5-06 

rheel',  D 

oi 

a 

4 

207 

16    51 

50 

3    11! 

7-27 

3- 

7-27 

=ill.       E 

I 

21 

01 

3 

a  Oi 

22 

12-9 

1      51 

400 

-949 

12-65 

■31,       F 

1 

2i 

4J 

fli 

3 

6    01 

IflJ 

1313 

45-0 

■949 

U-2S 

1 

j^R 

M 

3i 

6 

96 

IB 

8    0 

42 

4332 

3      6 

2-5 

7-95 

6-625 

■ 

BTh' 

46 

3 

8 

132 

17i 

54 

50 

1-7 

11- 

8-2 

■ 

KPl 

93 

3 

G 

lie 

10 

8  10 

1-87 

8-78 

5-47 

■ 

■VTk 

U 

3 

5 

64 

25 

5    1 

29 

55 

2      4 

3-57 

6-65 

7-91 

HPK'L 

QO 

2\ 

5 

90 

18 

A  II 

38 

42-63 

2      7 

2-5 

6-57 

4-64 

SSTm 

10 

2i 

d| 

77 

25 

*  n 

40 

48-5 

2      41 

5-75 

6-2 

11-88 

Btio-.     N 

6 

2| 

51 

60 

28 

3    7 

27 

1       71 

8-75 

5-25 

15-31 

Enrf.    O 

4 

21 

4: 

48 

32 

•2   10 

25 

6111 

1       6 

11-87 

4'S 

18-99 

liito.      P 

2 

2 

41 

62 

3    6 

2 

375 

8  8 

t-t 

10 

li 

6 

77 

25 

a  10 

40 

48'5 

1  in 

6- 

5- 

10 

12 

a 

66 

44 

2    8 

48 

60-5 

1    9 

25 

5-99 

4-95 

■ 

?/Slf  ™"i..  ^^l  "^i  " -"  fpu-a  fttar  .»  a«ro-  to.  ,h.  ..™u>  ..  it  i.  .«rin,  much 

•  »«oribiw'h«itliilheuioeintu'    """    "             i"""          emin.it 

«ia«  ID  thl.  MiinB,  whirh  hu  b»ii  l«  y«n  u  •ork,  U  ibe  wioi  of  binilih  in  th«  ipui-«h» 
ter  duMt  ln£ivi bHffl  6 ItHdM ot  mote, » Oic»  will  nol  ImI  h^ « long  «  Uie  IwveI— liMi,  uid 

M  hu  mndiD  imn .  u>d  hn  Docd'  WDriilpg  fur  lhi»  y«T>  pul. 

AlMlMHiramxiuibcKiolfiit 

^^H 

»t*!**TIOK  (IF  THE  TABLE  OF  WHBEI.S  IN  ACTUAL  USB  IN  MILLWORK. 

1 

!he  wheels  arc  all  reduced  to  what  may  be  called  one              ^^| 

nniDatioD.                                                                                   ^^H 

Irst — By  proportioning  their  breadths  all  to  what  they             ^^| 

lid  be  to  have  the  same  stTeng;th,  if  the  resistances             ^^| 

B  equal  to  tl>e  work  of  a  steam  engine  of  ten  horses*             ^^H 

^^1 

(condly — By  supposing  their  pitch  lines  all  brought  to             ^^H 

Mne  velocity  of  three  feet  per  second,  and  proportion-           ^^M 

96 


ON   THE   TEETH   OF   WHEELS.  [CHAP.   IV. 


ing  their  breadths  accordingly.  I  have  chosen  this  parti- 
cular velocity  of  3  feet  per  second,  because  it  is  the  velocity 
very  common  for  overshot  water  wheels. 

Such  cases  as  appear  to  have  worn  too  rapidly,  are 
marked,  which  may  tend  to  discover  the  limit  in  point  of 
breadth. 

Column    I  contains  the  horses'  power. 
2 .     .    .     .    pitch  in  inches. 

3  .     .     •     .     breadth  of  teeth  in  inches. 

4  .     .     •     •     number  of  teeth  of  wheeL 

5  .     .     .     .     revolutions  of  wheel  per  minute. 
6 .     .     .     .    diameter  of  wheel. 

7  •     .     .     •     number  of  teeth  of  pinion. 

8  .     .     .     •     revolution  of  pinion  per  minute. 

9  •     •     •     «     diameter  of  pinion. 

10  ...     .     breadth  proportionate  to  10  horses' 

power,  and  at  the  present  ve- 
locity. 

11.  .     .     .     present  velocity  per  second  in  feet 

12 .  .     .     .     breadth    in    inches   proportionate 

to  10  horses*  power,  at  3  feet 
per  second.  That  is,  all  the 
cases  reduced  to  the  same  de- 
nomination. 


OBSERVATIONS  ON  THE  TABLE  OF  WHEELS  IN  ACTUAL  USE  IN  MILL  WORK. 


121.  Having  reduced  the  examples  in  the  table,  in  the 
manner  already  described,  to  one  denomination,  the  results 
approach  nearer,  considering  all  circumstances,  than  could 
have  been  expected. 

1st.  In  two  of  the  cases,  viz.  b  and  d,  it  appears,  that 
the  wheels  were  rather  too  narrow  for  their  work.  These, 
however,  have  been  working  about  sixteen  years,  and  may 


I  CirAP.  IV.3  ON    THE    TEETH    OF    WHEELS. 


97 


[  jet  continue  for  a  long  time. — u  is  ■t'4'89  inches  in  breadth, 
I  then  reduced  to  10  liorses'  power,  at  3  feet  per  second. — 
D  is  7'^  inches  in  breadth.     The  pitch  of  both  is  three 
I inches. 

Three  inches  being  a  pitch  in  ver\'  general  use  for  the 
first  motion  of  mills,  could  the  proper  breadth  for  this  pitch 
be  ascertained,  it  would  servo  as  a  verj"  useful  standard. 

Rules,  to  be  of  practical  use,  must  be  easy  of  remem- 
brance, as  well  as  easy  of  application. 

Let  us,  therefore,  assume  a  simple  standard,  and  try 
liow  far  it  will  bear  the  test  of  esperience ;  for,  as  Du 
Buat  justly  observes,  "  It  is  an  excellent  method,  in  the  re- 
search of  obscure  difficult  truths,  to  suppose  a  theory  pre- 
existent,  founded  upon  the  most  probable  principles,  from 
which  may  be  determined,  the  choice  of  some  direct  ex- 
periments, proper  to  evince  the  fallacy  or  accuracy  of  the 
principles  proposed." 

The  actual  cases  in  the  table  may  be  considered  as 
satisfactory  experiments.  Let  us,  therefore,  try  the  fol- 
lowing simple  rule,  and  compare  some  of  its  results  with 
those  cases. 

Rule  I.  for  pitch  of  three  inches,  when  the  velocity  is 

tree  feet  per  second,  at  the  pitch  line. 
Make  the  teeth  as  many  inches  broad  as  the  number  of 
rsesi'  power  which  it  has  to  resist. 

For  example,  for  nine  horses'  power,  make  the  teeth 
ne  inches  broad. 

|ld&  Taking  this  example  then  as  a  point  from  which 

1  set  off,  and  supposing  the  same  breadth  of  nine  inches 

istanl,  we  shall,  in  the  first  following  table,  state  various 

tches,  and  (by  Proposition   I.)   shall  first  square  these 

iches,  to  find  the  number  of  horses'  power,  equal  to  the 

igth  and  durability  of  the  pitch,  when  the  teeth  are  all 

;  length.     The  strength,  thus  found,  will  be  inserted 


98  ON   THE   TEETH   OF  WHEEL8.  [CHAP.  IV. 

■ 

But,  as  the  lengths  generally  vary  as  the  pitches,  taking 
the  same  point  (three  inches  pitch,  nine  inches  hroad) 
from  which  to  set  off,  we  shall  diminish  the  value  of  the 
strength,  as  we  ascend,  and  increase  as  we  descend,  agree- 
ahly  to  Proposition  II.  The  results  will  be  found  in 
column  z. 

But  as  in  this  investigation  durability  is  of  equal  import- 
ance with  strength,  perhaps  the  true  proportion  may  be 
somewhere  between  the  results  in  the  columns  y  and  z. 


DESCRIPTION   OF  THE  SIX  FOLLOWING  TABLES  OF  PITCHES. 

In  all  the  following  tables,  the  column  w  contains  the 
pitch  in  inches. 

Column  X  contains  the  breadth  of  the  teeth  also  in 
inches. 

Column  Y  is  formed  upon  the  supposition,  that  the  teeth 
of  all  the  pitches  were  of  the  same  length,  and  contains  the 
strength  and  durability  of  the  teeth  valued  in  horses^ 
power. 

Column  z  contains  the  strength  and  durability  of  the 
teeth,  also  valued  in  horses'  power,  upon  the  supposition, 
that  the  lengths  of  the  teeth  are  in  the  same  proportion  to 
one  another,  as  the  pitches ;  and  that  the  strength  (by 
Prop.  II.)  is  inversely  as  the  length.  Having  taken  a 
three  inch  pitch  as  our  standard,  the  two  last  colunms,  Y 
and  z,  exactly  coincide  for  that  pitch ;  the  column  z,  de- 
creasing upwards  from  that  point,  and  increasing  down- 
wards in  an  inverse  ratio  of  the  lengths. 

It  is  evident,  that  the  tables  upon  these  principles  al- 
ready laid  down,  might  be  greatly  extended.  But  it  is 
hoped,  that  these  will  be  sufficient  for  our  present  purpose. 


CHAP.  IV.3  ON  THE   TBETH   OF  WHEELS. 


99 


I.   TABLB  OF  PITCHES 


123.  The  velocity  of  the  pitch  line  being  three  feet  per 
second,  and  the  breadth  of  the  teeth  nine  inches. 


w 

Y 

Z 

Pitch  in 
inches. 

Value  of 

Value  of 

stren^  in 

strength  in 

hones*  power. 

horses*  power. 

4 

16- 

12- 

34 

12-25 

10-5 

3 

9- 

9- 

2i 

6-65 

7-5 

2 

4- 

6- 

H 

2-25 

4-5 

1 

1- 

3- 

Supposing  again,  that  the  pitches  were  the  same  as  in  the 
first  table,  and  that  the  breadths  were  made,  in  each  par* 
ticular  case,  just  double  the  pitch,  then  the  horses'  power 
would  vary  as  in  the  following  table,  which  is  calculated 
by  taking  the  above  table,  and  by  direct  proportion,  finding 
the  horses'  power  equal  to  each  particular  breadth,  when  the 
breadth  is  just  double  the  pitch. 


II.   TABLE  OF  PITCHES 


124.  The  velocity  being  three  feet  per  second,  and  the 
breadth  of  the  teeth  double  each  pitch. 


w 

X 

Y 

Z 

Pitdiin 
inches. 

Twice  the 
pitch  in 
breadth. 

Value  of 

strength  in 

horses*  power. 

Value  of 

strength  in 

horses*  power. 

4 
3 
2 
1 

8 
7 
6 
5 
4 
3 
2 

14-22 
9-53 
6- 

3-47 

1-77 

•75 

•22 

10-66 

8-17 
6- 

4-16 
2-65 
1-5 
'66 

H  2 


100 


ON   THE   TEETH    OF   WHEELS.  [CHAP.  IV. 


The  strength  being  directly  as  the  breadth,  (by  Prop.  I.) 
it  is  easy  from  this  to  find,  by  the  Rule  of  Three  direct, 
the  horses'  power  equal  to  any  given  breadth  of  these 
pitches. 

The  stress  being  inversely  as  the  velocity,  (by  Prop.  II.) 
the  horses'  power  equal  to  any  of  these  pitches,  and  breadths, 
may  be  easily  found  for  any  other  velocity. 

But,  perhaps,  it  will  be  of  advantage  to  take  a  different 
view  of  the  subject,  by  fixing  upon  a  case  in  the  table  of 
wheels  in  actual  use,  and  proportioning  breadths  and 
pitches  from  it  for  various  velocities,  resistances,  &c. 

For  this  purpose  we  shall  select  the  case  h,  erected  by 
Boulton  and  Watt,  being  a  pitch  of  three  inches,  the  teeth 
8  inches  broad,  moving  with  a  velocity  of  11  feet  per 
second,  and  having  a  resistance  valued  at  the  power  of  46 
horses. 

Then  by  following  a  similar  procedure,  with  the  two 
foregoing  tables,  we  have  the  Tables  III.  and  IV.  propor- 
tionate to  the  case  h  at  a  velocity  of  1 1  feet  per  second. 


III. — TABLE   OP   PITCHES 


125.  Proportionate  to  h  in  the  Table  of  Wheels.  The 
breadth  of  teeth  (8  inches)  and  velocity  (eleven  feet  per 
second)  being  constant. 


w 

Y 

Z 

Pitrh  in 
inches. 

Hones*  power. 

Value  of 

strength  in 

horses*  power. 

4 

3i 
3 

2i 
2 

1 

81-77 
6261 
46- 
31-94 
20-44 
11-5 
511 

61-33 

53-66 

46- 

38-33 

30-66 

23- 

15-33 

CHAP,  iy.3  ON   THE   TEETH    OF   WHEELS. 


101 


IV. — TABLE  OP  PITCHES 


126.  Proportionate  to  h,  the  breadth  being  in  this  case 
double  the  pitch.  The  velocity  being  constant,  (eleven  feet 
per  second,)  but  the  breadths  double. 


w 

X 

Y 

Z 

Pitch  in 
inches. 

Breadthfl 

double 

the  pitch  in 

inches. 

Hones'  power. 

Value  of 

strength  in 

horses*  power. 

4 

H 
3 

2i 
2 

IJ 
1 

8 

7 
6 
5 
4 
3 
2 

81-77 
54-78 
34-5 
19-95 
10-22 
4-31 
1-28 

61-33 
46-95 
34-5 
23*94 
15-33 
8-62 
3-84 

But  it  may  be  satisfactory,  in  order  to  compare  with  the 
tables,  first  and  second,  to  reduce  the  velocity  to  three 
feet  per  second. 

The  two  following  tables,  therefore,  are  calculated  ac- 
cordingly at  that  velocity. 


v. — TABLE   OP   PITCHES 


127.  Proportionate  to  h,  at  a  velocity  of  three  feet  per 
second ;  the  breadth  being  constantly  eight  inches. 


w 

Y 

Z      . 

1^'x      1       • 

Value  of 

Pitch  m 
inches. 

Horses*  power. 

strenia^h  in 
horses'  power. 

4 

22-30 

16-72 

H 

17-07 

14-63 

3 

12-54 

12-54 

2i 

8-71 

10-45 

2 

5-57 

8-36 

li 

3-13 

6-26 

1 

1-39 

4-17 

102 


ON  THE   T£fiTH  OF  ¥mE£L8. 


[chap. 


VI. — ^TABLB  OP  PITCHES 


128.  Proportionate  to  h,  at  a  velocity  of  tliree  feet;  tb.e 
breadths  being  double  each  pitch. 


w 

X 

Y 

Z 

Pitch  in 
incfaei. 

Brcsutn  of 

teeth 
in  inches. 

Hones*  power. 

Value  of 

alrength  in 

hones*  povrer. 

4 

3 

2i 

2 

1 

8 

7 
6 
5 
4 
3 
2 

22-30 
14-93 
9-4 
5-44 
2-78 
117 
0-34 

16-72 
12-79 
9-4 
6-53 

4-17 
2-34 
1-02 

From  a  comparison  of  the  second  table  with  the  sixth, 
it  appears,  that  the  rule  we  have  annexed,  is  at  least  safe 
in  point  of  strength ;  for  by  that  rule,  a  three  inch  pitchy 
sis  inches  broadj  is  equal  to  a  strain  of  six  horses.  Whereas 
in  the  sixth  table,  the  same  pitch  and  velocity,  at  six  inches 
breadth,  has  strength  valued  at  nine  and  nearly  a  half 
horses*  power. 

But  when  we  consider  how  much  more  liable,  from  sand, 
&c.,  teeth  attached  to  water  wheels  are  to  wear,  than  those 
which  are  properly  greased  and  free  from  sand,  the  results 
correspond  as  nearly  as  could  be  expected. 

129*  Rule  II.  So  that,  taking  h  as  a  standard,  we  may 
conclude,  that,  for  a  pitch  of  three  inches^  with  a  velocity 
of  three  feet  per  second,  every  inch  of  breadth  may  be 
valued  at  one  and  a  half  horses*  power. 

The  first  rule  (Art.  121.)  I  think  therefore  may  safely  be 
followed  for  teeth  attached  to  water  wheels,  and  the  above 
conclusion  for  wheels  in  all  situations  where  they  are  pro- 
perly greased  and  free  from  sand. 

The  conclusions  here  drawn,  wiU,  I  think,  give  teeth 


CHAP.  IV.]  ON    THE    TEETH    OF    WHEELS.  103 

fufficiently  durable  and  strong  for  the  work  whi(.'h  they 

may  have  to  perform.     This  first  conclusion  gives,  perhaps, 

loo  great  a  result ;  how  much  may  with  prudence  be  de- 

I  ducted  from  the  results  of  either,  those  of  experience  will 

rietemiine.     But  to  the  young  millwright,  I  would  advise, 

I  of  the  two  extremes,  rather  to  err  in  makiug  his  work  too 

slrong.     Durability  ought  not  for  a  moment  to  bo  out  of 

light  in  the  arrangements  of  wheelwork  ;  there  are  many 

parts  of  machines  subjected  to  greater  stress,  but  not  liable 

to  wear;  whereas  the  t«eth  of  wheels,  the  moment  they 

begin  to  act,  begin  to  change  from  their  original  form,  and 

to  become  progressively  less  strong. 

130.  In  millwork,  at  present,  the  breadth  of  the  teeth, 
as  commonly  executed  by  the  best  masters,  seems  to  be 
from  about  twice  to  thrice  the  pitch. 

It  is,  perhaps,  not  easy  to  determine  what  proiwrtion  is 
on  the  whole  the  most  advantageous  for  the  breadth  of 

Eth.  A  fine  pitch,  on  the  one  hand,  gives  a  smooth  mo- 
1,  and  the  teeth  will  rub  less  on  each  other ;  but  an  in- 
aso  of  breadth  increases  in  some  degree  the  friction*. 
'the  durability,  as  well  as  the  strength  of  teeth,  is  per- 
haps nearly  in  direct  proportion  to  their  breadth. 

tlSl.  After  sending  the  foregoing  "  Jnqiiirif  respecting 
\e  Strength  of  the  Teeth  itf  JVheels  "  to  the  press,  Mr. 
ohn  Robcrton,  engineer,  perused  a  manuscript  copy  of  it, 
and  was  so  obliging  as  to  communicate  to  me  the  substance 
^of  what  follows.     His  rule,  it  will  be  readily  perceived,  is 
Hbnnded  on  the  principles  laid  down  in  the  "Inquiry;" 
Hnt  it  is  more  simple,  and  perhaps  more  accurate  than  the 
mode  of  approximation  which  occurred  to  me.     It  is  how- 
I  mt  satisfactory  to  find,  that  the  table  formed  on  his  rule, 
^'hich  from  his  experience  he  is  of  opinion  cannot  be  far 
the  truth,)  very  nearly  coincides  in  its  results  with 
pies  fifth  and  sixth.     It  may  be  observed,  that  he  founds 
*  See  Art.  C7  and  68. 


104  ON  THE  TEETH  OF  WHEELS.     [CHAP.  IV. 

his  calculations  upon  the  thickness  of  the  teeih^  wluch  in 
all  cases  he  supposes  a  little  less  than  half  the  pitch,  which 
proportion  is  very  common  in  practice.  When  hoth  wheel 
and  pinion,  however,  are  of  cast  iron,  it  is  evident,  that, 
heing  more  liahle  to  wear,  the  teeth  of  the  pinion  ought  to 
he  thicker  than  those  of  the  wheel. 


CONSTRUCTION   OP  THE  FOLLOWING  TABLB. 

132.  The  thickness  of  the  teethj  in  each  of  the  lines,  is 
varied  one-tenth  of  an  inch.  The  breadth  of  the  teeth  is 
always  four  times  as  much  as  their  thickness.  The  strength 
of  the  teeth  is  ascertained  by  multiplying  the  square  of 
their  thickness  into  their  breadth^  taken  in  inches  and 
tenths,  &c.  The  pitch  is  found  by  multiplying  the  thick- 
ness of  the  teeth  fry  2*1.  The  number  that  represents  the 
strength  of  the  teeth,  will  also  represent  the  number  of 
horses*  power,  at  a  velocity  of  about  four  feet  per  second. 
Thus  in  the  table  where  the  pitch  isS'\5  inches^  the  thick- 
ness  of  the  teeth  1*5  inches^  and  the  breadth  6*  inches^  the 
strength  is  valued  at  13^  horsed  power ^  with  a  velocity  of 
four  feet  per  second  at  the  pitch  line. 


rCH-tP.  IV.]  ON    THE   TEETH    OF    WHEELS. 


k  3ABLB  OP   PITCHES  OP  WHEELS, 


13S.  With  the  breadth  and  thickness  of  the  teeth,  and 
Ijhe  correBponding  number  of  horses'  power,  moying  at  the 
*  pitch  line  at  the  rate  of  three  feet,  of  four  feet,  of  six  feet, 
and  of  eight  feet  per  second. 


f 


Strength  of 

iTiict- 

leeOi,  or 

HorWB- 

Honun' 

Pbchb 

nsBof 

BrettJih 

Hones'  poser 
at  three  feet 
pCTBccond. 

poneral 

power  at 

iulHl. 

teetbm 
incha. 

ininSes. 

J^foTfra'^r 

I^^Di 

wcond. 

'    3'fl9 

1-il 

7-fl 

27-43 

20-57 

4114 

54-85 

.  n$ 

1-8 

7-2 

23 

32 

17-49 

34-98 

46-64 

1  a-37 

1-7 

6-8 

19 

65 

14-73 

29-4G 

39-28 

1    3'36 

le 

e-4 

16 

38 

12-28 

2i-,5G 

32-74 

1    3-13 

1-5 

e- 

13 

5 

10-12 

20-24 

26-98 

2-8* 

1-4 

5-e 

10 

97 

8-22 

16-44 

21-92 

'    2-73 

13 

5-2 

8 

78 

6-58 

13-16 

17-54 

J-52 

1-2 

+■8 

fi 

91 

5-18 

10-36 

13-81 

S-31 

1-1 

*1 

5 

32 

3-99 

7-98 

10-64 

21 

!-0 

4- 

4 

0 

3-0 

B'O 

8-0 

i-8a 

■iJ 

3-6 

S 

91 

2-18 

4-36 

5-81 

I-fi8 

■s 

3-2 

2 

04 

1-53 

3-06 

3-08 

1-47 

-7 

2-8 

1 

37 

1.027 

2-04 

2-72 

\u 

-6 

2'4 

86 

■64 

1-38 

1-84 

1-05 

■5 

2- 

■5 

•375 

-75 

r 

James  Carmichael,  millwright,  (of  Dundee.^  made  the 

following-  remarks  to  Buchanan  on  the  strength,  &c.,  of 

wheelwork. 

ilS-t.  "  Sir — It  is  a  corroboration  of  the  truth  of  the 

iles  of  pitches,  that  Mr.  Roberton's  table  coincides  very 

ly  with  columns  marked   y  in  jour  tables ;  but  he 

to  have    overlooked   the    propriety  of  taking    the 

:h  of  the  teeth  into  bis  calculations.     I  am,  therefore, 

gtill  of    opinion,  that  the  true  value  is  in  the  columns 

aiarked  z  in  your  tables. 

Admitting  the  truth  of  the  fundamental  propositions. 


106 


ON   THE   TEETH  OF  WII&&L8.  j^CfiAP;  IT^ 


and,  from  a  comparison  of  the  tables  of  pitches,  I  would 
propose  the  following  rule,  which  is  on  the  same  principle 
as  that  of  columns  z  in  your  tables,  for  calculating  the 
proportionate  strength  of  the  teeth  of  wheels. 

"  Rule. — Multiply  the  breadth  of  the  teeth  by  the  square 
of  the  thickness,  and  divide  the  product  by  the  length. 
The  quotient  will  be  the  proportionate  strength  in  horses* 
power,  with  a  velocity  of  2*27  feet  per  second. 

'*  By  that  rule  I  have  calculated  the  following  table ; 
and,  for  the  sake  of  comparison,  I  have  taken  three  cases 
from  Mr.  Roberton's  table,  and  three  from  your  Tables  3d 
and  5th. 


(b 


EXPLANATION   OF  THE   TABLE. 


"  Column  2  contains  the  thickness  of  the  teeth.  The 
pitch  is  found  by  multiplying  the  thickness  by  2'1  * ;  and 
the  length  is  found  by  multiplying  the  thickness  by  l*2t. 

"  Column  5  contains  the  proportionate  strength,  and 
also  the  number  of  horses*  power  (proportionate  to  the  case 
H,  see  p.  100)  which  the  teeth  are  equal  to,  with  a  velocity 
of  2*27  feet  per  second. 


1 

2 

3 

4 

5 

6 

7 

8 

Strength  of 
teeth,  or 

number  of 

horses* 

power,  at 

2-27  feet 

Pitch  in 
inches. 

Thick- 
ness of 
teeth  in 
inches. 

Breadth 

of  teeth 

in  inches. 

Length 

of  teeth 

in  inches. 

Horses* 

power  at 

three  feet 

per  second. 

Horses* 

power  at 

SIX  feet  per 

second. 

Horses* 

power  at 

eleven  feet 

per  second. 

per  second. 

3-9 

1-9 

7-6 

2-28 

11-73 

15-46 

30-92 

56-84 

2-9 

1-4. 

56 

1-68 

6-53 

-    8-63 

17-26 

31-64 

2-1 

1- 

4- 

1-2 

3-33 

4-4 

8-8 

16-1 

4- 

1-904 

8- 

2-285 

12-698 

16-78 

33-56 

61-52 

3- 

1-428 

8- 

1-714 

9-523 

12-58 

25-16 

46- 

IJ 

•714 

8- 

•857 

4-752 

6-27 

12-54 

23-02 

*  That  is  in  order  to  make  the  space  between  the  teeth  a  little  wider 
than  the  thickness  of  a  tooth.     See  Art.  132, 
t  Respecting  the  length  of  teeth,  see  Art.  145. 


cn.ip.  IV.] 


ON    THE    TEETH    OF    WHEELS. 


107 


"  REMARKS. 

"1st.  The  last  three  cases  in  the  table  are  taken  from 
the  Tables  3d  and  5tb,  and  the  results  coincide  so  well 
with  the  columns  marked  z,  that  I  presume  the  rule  is 
just. 

"  2dly.  The  first  three  cases  are  from  Roberton's  table  ; 
ihe  second  case  is  very  near  the  same  as  in  his  table  -,  but 
the  first  is  considerably  less,  and  the  third  considerably 
more.  Hence  I  infer  that  Roberton  has  taken  his  data 
from  a  pitch  about  three  inches. 

"  3dly.  If  any  two  wheels  have  the  length  and  thickness 
of  their  teeth  in  the  same  proportion  to  their  respective 
pitches,  the  breadth  of  the  teeth  and  the  velocity  being 
the  same,  i/ie  strength  will  be  dirextli/  as  the  pitches.  The 
truth  of  this  is  deduced  from  the  columns  marked  z  in 
Tables  1st,  3d,  and  5th." 

135.  Table  of  pitches  of  wheelwork,  with  the  brendtk 
and  thickness  of  the  teeth,  and  the  corresponding  strength 
io  horses*  power,  calculated  by  Carmichael's  rule. 


TtWt- 

Breadth 

Lennh 
oftoHh 
LDiDches 

Sireneth  of 

Hor-e,' 

Hoi««' 

Hor««' 

!>iidib 

teeth  in 

uftedh 
iaiDchis. 

tW  fe^l 

power  al 
iii  feet 

power  at 
clcvpti  fret 

inchea. 

p«f  second. 

[ler  lecond. 

3-90 

!•» 

7-0 

2-28 

12-03 

15-90 

31-80 

58-30 

■^■78 

1-8 

7-2 

2-16 

10-80 

U-27 

28-5* 

52-32 

3-57 

1-7 

6-8 

20* 

9-63 

12-72 

.     25'54 

46-68 

3-36 

1-6 

6-4 

1-92 

8-53 

11-27 

32-54 

41-32 

315 

1-5 

6-0 

1-8U 

7-50 

9-91 

19-82 

3G-33 

2»* 

1-4 

5-e 

l-(58 

e-53 

8-G3 

17-26 

31-64 

2-73 

1-3 

52 

l-.')6 

5-63 

7-44 

14-88 

27-28 

S-52 

1-2 

4.-8 

1-+4 

4-80 

G-34 

12-C8 

23-24 

S-31 

1-1 

+■4 

1-32 

+■03 

5-32 

1 0-84 

1954 

S-10 

l-« 

40 

1-20 

3-33 

4-40 

6-81 

ie-15 

1-89 

0-3 

3-6 

1-08 

2-70 

3-57 

7-1* 

13-09 

<  l-M 

(1-8 

3-2 

0-96 

2-13 

2-81 

5-62 

10-33 

,3-»" 

0-7 

2-8 

0-8* 

l-«3 

215 

4-30 

7-88 

1-86 

0-« 

8-4 

0-72 

1-20 

1-.S9 

3-18 

3-83 

:  1-05 

0-5 

2-0 

O'liO 

0-83 

110 

220 

4-03 

108 


ON   THE   TEETH   OF'  WHEELS.  [CHAP.  IV. 


136.  It  is  not  perhaps  quite  so  difficult,  as  our  author 
imagined,  to  determine,  from  first  principles,  the  strength 
proper  for  teeth  of  wheels,  and  such  a  method  must  always 
be  preferred  to  empirical  rules.  We  sh|dl  here  show  how  to 
apply  those  principles  which  will  give  the  reader  an  oppor- 
tunity of  comparing  the  two  methods. 

In  the  first  place  let  us  consider  under  what  circum- 
stances the  strain  on  a  tooth  will  be  the  greatest  possible- 
Let  A  B  c  D  be  the  side  of  a  tooth,  then  it  will  be  evident, 
that  the  strain  will  be  greatest,  when  the  stress  is  thrown 


upon  one  comer  of  the  tooth,  as  at  c,  whether  it  be  from 
irregular  action  or  from  any  substance  getting  between  the 
teeth. 

In  such  a  case,  it  may  be  shown  by  the  rules  of  mcurima 
and  minimaj  that  e  c  being  equal  to  c  b,  the  strain  will  be 
greatest  in  the  line  e  b  ;  and,  in  the  case  of  fracture,  it 
would  take  place  according  to  that  line. 

Since  the  thickness  of  a  tooth  is  not  regular,  we  shall 
have  a  result,  sufficiently  near  for  this  purpose,  if  we 
express   the   relation  between   the   stress   and  strain  by 

^ =-^ ^  ^^  ^  ^ — .      (Essay  on    Cast   Iron.   Art 

V  8(fc)  ^        ^ 

81,  and  Art.  107  of  this  Essay,)  which,  wheny  = 
15,300  lbs.  on  a  square  inch,  reduces  to  — ZJi=rf*  = 


V 


the  square  of  the  thickness  of  the  tooth  in  inches.  But,  a 
tooth  should  be  capable  of  resisting  this  stress,  when  it  is  con- 
siderably worn  by  friction ;  and  an  allowance  fully  equal  to 


CHAP.   IV.]  ON    THE    TEETH    OF    WHEELS.  109 

that  which  ought  to  take  place  before  renewing  the  wearing 

parts  of  the  machine,  will  be  one  third  of  the  thickness  of  a 

tooth.    Now  to  allow  of  this  degree  of  wear  in  the  tooth,  and 

that  it  shoidd  remain  equal  to  the  stress,  it  may  be  easily 

shown  that  the  tooth  should  be  capable  of  resisting  2^  times 

.  ,      ^  ,       A       ■■''5G  H         „         S     H      , 

(he  power  at  the  first ;  therefore  — —  =  d" :  or  7  ^/  -  =  "■ 

^\Tiere  h  is  the  number  of  horses  which  are  equal  to  the 
poirer  of  the  first  mover ;  v  the  velocity  of  the  pitch  line 
of  the  wheel  in  feet  per  second,  and  d  the  thickness  of  a 
looth  in  inches. 

Tliis  investigation  furnishes  an  easy  practical  rule  for 
the  thickness  of  teeth  ;  and,  consequently,  for  the  pitch  of 
wheels  and  pinions ;  we  shall  give  it  in  words  at  length, 
nith  an  example,  and  then  proceed  to  determine  a  rule  for 
the  breadth  of  teeth. 


137.  Find  the  number  of  horses  which  are  equivalent 
to  the  power  of  the  first  mover  of  the  train  of  machinery, 
and  divide  that  number  by  the  velocity,  in  feet  per  second, 
of  the  pitch  line  of  the  pinion  or  wheel ;  extract  the  square 
root  of  the  quotient,  and  three  fourths  of  this  root  will  be 
the  least  thickness  of  the  tooth  for  the  wheel  or  pinion,  in 
laches. 


1    LEAST    QUA.VTril-    OF    PITCH    FOR    A    WHEEL    OB    P 


^Hl38.  If  the  thickness  of  the  teeth  of  the  pinion  be  in- 
^tonded  to  be  the  same  as  those  of  the  wheel,  multiply  the 

:)ili.-kncse  above  determined  by  2*1,  the  product  will  be  the 

Tiiich  required. 
^HBut  wc  may  observe,  that  if  a  pinion  makes  three  turns. 


110  ON   THE  TEETH   OF  WHEELS.  [CHAP.  IV. 

for  example,  while  the  wheel  makes  one,  the  teeth  of  the 
pinion  will  be  worn  three  times  the  quantity  of  those  of  the 
wheeL  Hence,  to  provide  against  such  excess  of  wear,  if 
the  pinion  makes  n  revolutions,  while  the  wheel  makes 

one,  the  pitch  should  be d  inches,  and  the  thickness 

3 

of  the  teeth  of  the  pinion d  inches,  when  d  is  the 

thickness  of  the  teeth  of  the  wheeL 


EXAMPLE. 

139.  Let  the  force  of  the  first  mover  be  equivalent  to 
ten  horses,  and  the  velocity  of  the  pitch  line  three  feet  per 
second.  Dividing  10  by  3,  we  have  33- ;  and  the  square 
root  of  3^  (by  the  Table  of  Powers,  Art  479.)  is  1-83 
nearly;  and  f  x  1*83  =  1*45  inches  for  the  thickness  of 
the  teeth  of  the  wheel. 

Again,  suppose  the  pinion  to  turn  twice  while  the  wheel 

turns  once,  then  n  =  2,  and  x  1*45  =  3*5  inches 

3 

the  pitch. 
And X  1  '45  =  1  "93  inches,  the  thickness  of  the 

teeth  of  the  pinion. 


THICKNESS  OP  WOODEN   TEETH. 


140.  The  kind  of  wood  employed  for  teeth,  is  usually 
about  one  fourth  of  the  strength  of  cast  iron,  and  since 
the  thickness  of  the  teeth  should  vary  inversely  as  the 
square  root  of  the  power  of  the  material,  the  square  root 
of  \  being  i,  wooden  teeth  should  be  twice  the  thickness 
of  cast  iron  teeth.  The  pitch  of  course  will  be  greater  in 
the  same  proportion. 


CHAP.   IV.]  ON    THE    TEETH    OF    WHEELS. 


TO  DSTERMINE  THE   BRBADTH   OF  CAST    IRON   TEETH. 

UI.  That  case  where  a  beam  is  fixed  at  one  end,  and 
thi-  load  acts  at  the  other,  applies  to  the  teeth  of  wheels, 

in  their  general  state  of  action,  and  the  stress  is  i -. 

fArt.   107.)       Ilcnce,  when  /    =   the  length,   and  b   = 

the  breadth  of  a  tooth,  T12iL'  =  212  b  d\     (Essar/  on 

Out  Iron,   Art.  116.)  And  to   allow  one  third  of  the 

thickness   of  the    tooth  for  wear,  the  equation  becomes 
.'I  j(  -j-.i^)  jj  I 

— =  212  b  d*.  But  we  have  already  seen,  that 

^=•££^1    therefore ?ijlli^^^= -556  b  x  212,    or 

H!=b. 

This  calculation  informs  us  what  breadth  is  essential 
tor  strength ;  that  is,  the  breadth  should  never  be  less 
ihan  1-2  multiplied  by  the  length  of  the  tooth;  but,  it 
may  be  proved  that  the  durability  is  nearly  in  direct  pro- 
portion to  the  breadth,  and  inversely  as  the  pressure. 
Some  of  the  maxims  of  our  author,  as  far  as  regards  the 
breadths  of  teeth,  agree  well  with  the  theory  of  durability  j 
but  the  conclusions  respecting  strength  and  the  limits  of 
pitch  are  not  so  much  to  be  relied  upon  ;  indeed  the  facts 
drawn  from  practical  construction  are  not  so  well  adapted 
for  tlie  latter  object. 

1 4-^2.  If  we  suppose  that  teeth  sis  inches  in  breadth  are 
idfficienfly  durable  for  a  power  equivalent  to   10  horses, 

en  the  pitch  line  moves  at  the  rate  of  3  feet  per  second  j 
the  Table  of  Wheels  in  p.  95,  Art.  120.  seems  to  indi- 
that  this  supposition  is  near  the  truth ;    therefore 

'"'     (■       "     /        3  X  6  X  H     1-8 H     TT,  .  •  I.,-  1    ..1. 

-  ;  o  :  :  -  :  o  =  = ^.     Ihat  is,  multiply  the 

■j  F  10  w  " 


HpBC^   po 


power  hy  X"8,  and  divide   by  the  velocity  of  the 


112  ON   THE   TEETH    OF   WHEELS.  [CHAP.  IV. 

pitch  line  in  feet  per  second,  the  quotient  will  be  the 
breadth  in  inches. 


EXAMPLE. 

Taking  the  case  h   in   the   table   Art.  120,  we  have 

= — =7'53  inches:  Messrs.  Boulton  and  Watt 

V  11 

in  this  case  made  the  breadth  8  inches. 

T    ,,  ,     .,  1         1'8h       1*8  X  14 

In  the  case  k,  by  the  same  makers, =  —pm = 

•^  V  6-65 

3'8  inches.     The  breadth  actually  employed  was  5  inches. 

Hence  it  appears,  that,  in  a  considerable  range  of  power 

and  velocity,  our  formula  gives  results  below  those  actually 

employed,  but  the  breadth  assigned  by  the  table  calculated 

by  Mr.  Roberton,  Art.  133,  is  always  vastly  below  ours  in 

the  greater  powers ;  indeed  it  is  manifestly  erroneous  in 

the  breadths,  for  where  the  moving  power  is  doubled,  the 

breadth  is  increased  only  one  third.     In  the  thickness  of 

teeth,  it  nearly  agrees  with  our  rule. 


BREADTH   OP   WOODEN    TEETH. 


143.  For  wooden  teeth  we  may  take,  as  the  basis  of  a 
practical  rule,  the  case  n  in  the  table,  Art.  120 ;  which,  ex- 
pressed in  the  nearest  whole  number,  is  —  =  6  in  inches. 

V 


EXAMPLE. 


In  the  case  o ;  h  =4  horses,  and  v  =4*8  feet  per  second ; 

therefore,  —  = =4*17  inches  the  breadth :  the  actual 

i;        4-8  ' 

breadth  used  was  4f  inches. 


,v.] 


ON    THE   TEETH    OK    WHEELS. 


113 


OP   THE  GIBENOTH   OF  aiAVE9   FOR  TBUNDLBS. 

144.  TTiis  is  a  subject  our  author  has  not  touched  upon, 
but  we  consider  it  necessary  to  examine  the  strength  of 
stares,  because  trundles  seem  capable  of  improvement,  and 
thi'v  have  some  advantages  which  toothed  pinions  have  not. 

If  tlie  length  of  a  stave  in  feet  ho  I,  its  diameter  in 

inches  «/,  and  the  stress  upon  it  ^ ;  which  is  supposed 

ti)  act  at  the  weakest  part  of  the  stave ;  that  is,  in  the 
middle  of  its  length ;  then,  by  the  rule  for  the  strength  of 

cjlmders,  (Sssai/  on  Cast  Iron',  Art.  129,) =rf^ 

Now,  if  it  be  made  to  resist  3^  times  the  power  when  first 

made,  in  order  to  allow  for  wear :  we  shall  have    *^  ' 

500  V 

=d',  or  ( )    =d;  or  with  sufficient  accuracy,  it  is 

^L  EXAMPLE. 

^r  Let  the  power  of  the  first  mover  be  equal  to  10  horses, 
the  velocitv  of  the  pitch  line  3  feet  per  second,   and  the 

rth  of  the  stave  -{i  of  a  foot ;  then  2  i^)  '  =  a  (l^ilA' )  * 
'  ^  V  '  ^     3      ' 

ix  1*442  =  2"884'  inches  the  diameter  of  the  stave. 

p45.  We  take  the  liberty  of  inserting  the  following  tablo 

I  a  respectable  periodical  publication,  as  we  purpose  to 

K  its  application,  knowing  it  maybe  of  use  to  millwrights  f- 

'  TredgoM'ii  work,  whicL  we  occasionally  quote. 

t  This  very  useful  table  was  printed  in  a  small  pamjihlet,  price  one 

I  1803,  but  IE  nt  present  out  of  print.     Wben  tbe  nuniber  of 

1  10,  look  for  tbe  nulius   of  double   tbe  number  of  tcetb, 

Pit  win  be  the  mdius  required.     Wbeu  the  number  of  teeth  ex- 

^look  for  the  radius  of  half  tbe  number  of  teeth,  the  double  of 

Kieh  win  Ik  tie  otie  required. 


114 


ON   THE   TEETH   OF  WHEELS.  [CHAP.ITb 


TABLE  OP  THE  RADII   OP   WHEELS,   PROM   TEN   TO    THRU    BUlHttlD  IIRI, 

THE   PITCH*  BEING  TWO  INCHES. 
BY    B.   DONKIN,   ESQ.,   CIVIL   ENGINEER,   LONMIT. 


No.  of 

Radius  in 

No.  of 

RadiuB  in 

No.  of 

Radius  in 

No.  of 

Rib. 

teeth. 

inches. 

teeth. 

inches. 

teeth. 

inches. 

teeth. 

•^^^L^ 

10 

3-236 

47 

14-972 

84 

26-741 

121 

88-5« 

11 

3-549 

48 

15-290 

85 

27063 

122 

88-881 

12 

3-864 

49 

15-608 

86 

27-381 

123 

39-151 

13 

4-179 

50 

15-926 

87 

27-699 

124 

39-47 

U 

4-494 

51 

16-244 

88 

28-017 

125 

89^ 

15 

4*810 

52 

16-562 

89 

28-336 

126 

40-11 

16 

5-126 

53 

16-880 

90 

28-654 

127 

40-4S 

17 

5-442 

54 

17-198 

91 

28-972 

128 

40-74 

18 

5-759 

55 

17-517 

92 

29-290 

129 

41-06 

19 

6-076 

56 

17-835 

93 

29-608 

130 

41-38 

20 

6-392 

57 

18-153 

94 

29-927 

131 

4f7a 

21 

6-710 

58 

18-471 

95 

30-245 

132 

42-OS 

22 

7-027 

59 

18-789 

96 

30-563 

133 

42-33 

23 

7-344 

60 

19-107 

97 

30-881 

134 

42-63 

24 

7-661 

61 

19-425 

98 

31-200 

135 

42-97 

25 

7-979 

62 

19-744 

99 

31-518 

136 

43-28 

2(J 

8-296 

63 

20-062 

100 

31-836 

137 

43-61 

27 

8-614 

64 

20-380 

101 

32-155 

138 

48-93 

28 

8-931 

65 

20-698 

102 

32-473 

139 

44-24 

29 

9-249 

66 

21-016 

103 

32-791 

140 

44-56 

30 

9-567 

67 

21-335 

104 

33-109 

141 

44-88 

31 

9-885 

68 

21-653 

105 

33-427 

142 

45-26 

32 

10-202 

69 

21-971 

106 

33-746 

143 

45-5S 

33 

10-520 

70 

22-289 

107 

34-064 

144 

45-84 

34 

10-838 

71 

22-607 

108 

34-382 

145 

46-15 

35 

11-156 

72 

22-926 

109 

34-700 

146 

46-47 

36 

11-474 

73 

23-244 

110 

35-018 

147 

46-79 

37 

11-792 

74 

23-562 

111 

35-337 

148 

47-11 

38 

12-110 

75 

23-880 

112 

35-655 

149 

47-43 

39 

12-428 

76 

24-198 

113 

35-974 

150 

47-75 

40 

12-746 

1  t 

24-517 

114 

36-292 

151 

48-06 

41 

1 3-064 

78 

24-835 

115 

36-61 1 

152 

48-38 

42 

13-382 

79 

25-153 

116 

36-929 

153 

48-70 

43 

13-700 

80 

25-471 

117 

37-247 

154 

49-08 

44 

14-018 

81 

25-790 

118 

37-565 

155 

49-34 

45 

14-336  1 

82 

26-108 

119 

37-883 

156 

49-66 

46 

14-654 

83 

26-426 

120 

38-202 

157 

49-97 

*  By  the  pitch  is  understood  the  distance  between  the  oentrM  of  H 
contiguous  teeth ;  and  by  the  radius  is  understood  the  distance  betwaea  t 
centre  of  the  wheel  and  the  centre  of  each  tooth.  For  any  other  pill 
say,  as  two  inches  is  to  the  radius  in  the  table,  so  is  the  gi^en  pitdi  fee  i 
radius  required. 


pp.  IV.]            ON    THE    TF,ETU    OF    WHEKLS.                              115                 ^^^| 

B    bdkub 

No.  of 

lUdiluiD 

No.  of 

lUdius  in 

No.  of 

Htulimin 

^^H 

■      iach«>. 

teeth. 

iMLh. 

inches. 

teeth. 

inchau 

1 

lig  1  ao-ane 

194 

61-755 

230 

73-214 

266 

84-673 

I!3       oO-ei5 

195 

62-073 

231 

73-532 

267 

84-901 

m       50933 

196 

62-392 

232 

73-850 

268 

83-300 

^^^^1 

1        51-25) 

197 

62-710 

233 

74-168 

260 

83-627 

^^^^H 

i       51-569 

198 

63-028 

234 

74-487 

270 

85-946 

^^^^H 

'(        51-888 

100 

63-346 

235 

74-805 

271 

86-264 

^^^^H 

H        52-20e 

200 

63-665 

236 

75-123 

272 

86-582 

^^^^1 

ffi        52-524 

201 

63-983 

237 

75-441 

273 

86-900 

^^^^1 

«B        52-8*3 

202 

64-301 

238 

75-700 

274 

87-219 

^^^H 

fi7        53-1  Gl 

203 

64-620 

230 

76-078 

275 

87-537 

■^^^B 

S8        fi3-i79 

204 

64-938 

240 

76-397 

276 

87-855 

^^^^M 

W        53-798 

205 

65-236 

241 

76-715 

277 

88-174 

^^^^M 

W        5VII8 

206 

63-574 

242 

77-1133 

278 

88-492 

1         5t-434 

207 

63-893 

243 

77-351 

279 

88-810 

^^^^M 

5*--52 

208 

6B-211 

244 

77-670 

280 

89-120 

^^^^M 

55-071 

209 

66-529 

245 

77-988 

281 

80-447 

^^^^M 

;         55-380 

210 

66-848 

246 

78-306 

282 

80-763 

^^^^M 

55-7fl7 

211 

67-166 

247 

78-623 

283 

90-084 

^^^^M 

56-026- 

212 

67-484 

248 

78-943 

284 

80-402 

^^^H 

5B344 

213 

67-803 

249 

79-261 

285 

90-720 

^^^H 

■-         58-662 

214 

68-121 

230 

70-580 

286 

91-038 

^^^^1 

P         56-980 

213 

68-439 

251 

70-898 

287 

91-357 

^^^H 

57-299 

216 

68-757 

252 

80-216 

288 

91-675 

^^^H 

57-617 

217 

69-075 

253 

80-534 

289 

91-993 

^^^^1 

57-935 

218 

69-394 

234 

80-853 

290 

92-312 

^^^^M 

58-253 

219 

69-712 

253 

81-171 

291 

92-630 

^^^^M 

58-572 

220 

70-031 

256 

81-489 

292 

92-948 

^^^^M 

58-890 

221 

70-349 

257 

81-808 

293 

93-267 

^^^^M 

5S-2O0 

222 

70-667 

258 

82-126 

294 

03-585 

^^^^M 

59-327 

223 

70-985 

259 

82-444 

295 

93-903 

59-845 

224 

71-304 

260 

82-763 

296 

04-222 

^^^H 

eo-163 

225 

71-622 

261 

83-081 

297 

94-540 

^^^H 

00-482 

22G 

71-941 

262 

83-399 

298 

94-858 

^^^H 

90-800 

227 

72-258 

263 

83-717 

299 

95-177 

^^^H 

6M18 

22S 

72-577 

264 

84-036 

300 

95-495 

^^^H 

6I-43S 

229 

72-805 

265 

84-354 

^^M 

OF  AO&ANGmC   THE   NUMBERS   OF   WHEEL-WORK.                                                  ^^^H 

,  In  a  machine,  the  velocity  of  the  impelled  pniat              ^^^| 

be  to  that  of  the  working  point  in  the  ratio  which              ^^H 

pted  to  the  maximuni  effect  of  the  moving  power  on              ^^^| 

le  port,  and  the  best  working  effect  on  the  other  part.              ^^^| 

ther  arrangement  of  the  relative  motions  of  the  parts             ^^H 

.    M 

116  ON    THE   TEETH    OF   WHEELS.  [cHAP.  IV. 

of  a  machine  must  clearly  be  attended  with  a  loss  of  power, 
or  the  work  will  not  be  done  properly.     But  when  the  best 
working  velocity  is  known,  and  also  that  which  enables  the 
first  mover  to  produce  the  greatest  effect ;  the  proper  ar- 
rangement 6f  the  numbers  of  the  teeth  of  the  wheels  and 
pinions  is  a  very  simple  operation.     The  subject  has  been 
treated  of  for  particular  machines,  by  several  writers ;  but 
since  it  has  been  chiefly  under  a  somewhat  erroneous  view 
of  the  real  nature  of  the  maximum  effect  of  machines,  it 
will  be  perhaps  of  use  to  give  a  general  formula,  and  a  few 
particular  examples,  to  save  the  trouble  of  reference,  and 
render  the  work  somewhat  more  complete*. 

It  will  be  an  advantage  to  advertise  the  young  mechanic 
of  one  or  two  essential  particulars,  before  proceeding  to  the 
principal  object. 

147.  In  the  first  place,  when  the  wheels  drive  the 
pinions,  the  number  of  teeth  in  any  one  pinion  should  not 
be  less  than  8 ;  but  rather  let  there  be  11  or  12  if  it  can 
be  done  conveniently.  And  in  the  particular  form  of  teeth 
described  in  Art.  30,  the  number  of  teeth  in  a  pinion 
should  not  be  less  than  10 ;  but  it  would  be  better  to  have 
13  or  14.     (See  Art.  34  to  37.) 

148.  Secondly,  when  the  pinions  drive  the  wheels,  the 
number  of  teeth  on  a  pinion  may  be  less ;  but  it  will  not 
in  anv  case  be  desirable  to  have  fewer  than  6  teeth  on  a 
pinion ;  and  give  the  preference  to  8  or  9,  where  it  can  be 
done  with  convenience. 

149.  Thirdly,  the  number  of  teeth  in  a  wheel  should 
be  prime  to  the  number  of  teeth  in  its  pinion  ;  that  is,  the 

*  The  methods  of  adjusting  the  numbers  of  wheel- work,  so  that  the  con- 
temporary revolutions  may  be  always  in  a  given  ratio,  is  a  distinct  branch 
of  this  subject,  chiefly  useful  in  clock  and  watch- work,  planetary  machines, 
and  the  like ;  and  since  our  plan  does  not  include  the  construction  of  such 
machines,  the  reader,  desirous  of  such  information,  may  consult  Camus  on 
the  Teeth  of  Wheels. 


CHAP.  IV.3  ON   THE   TEETH    OF   AVHEELS.  11? 

number  representing  the  teeth  in  the  wheel  should  not  be 
divisible  by  the  number  of  teeth  in  the  pinion  without  a 
remainder.  And  as  the  numbers  of  pinions  will  in  general 
be  first  settled,  it  will  be  an  advantage  to  take  a  prime 
number  for  each  pinion,  as  7>  11>  13,  175  19,  23,  &c.,  be- 
cause such  numbers  are  seldomer  factors  than  others. 
But  when  it  happens  that  a  prime  number  can  be  directly 
fixed  upon  for  the  wheel,  any  whole  number  which  ap- 
proaches near  to  the  required  ratio  will  answer  for  the 
pinion;  as  minute  accuracy  is  not  required.  A  prime 
number  for  the  wheel,  or  one  which  is  not  divisible  by  the 
number  of  the  pinion,  is  esteemed  the  best,  because  the 
same  teeth  will  not  always  come  together,  and  the  wear 
will  be  more  uniform. 

150.  Foiurthly,  if  it  be  desired  that  a  given  increase  or 
decrease  of  velocity  should  be  communicated  with  the  least 
quantity  of  wheel- work,  it  has  been  shown  that  the  number 
of  teeth  on  each  pinion  should  be  to  the  number  on  its 
wheel,  08  1  :  3-59.  (Dr.  Young's  Nat.  Phil.  Vol.  II.  Art. 
S66.)  But,  on  account  of  the  space  required  for  several 
wheels,  and  the  expense  of  them,  it  will  often  be  necessary 
to  have  5  or  6  times  the  number  of  teeth  on  the  wheel  that 
there  is  on  the  pinion.  The  ratio  of  1  :  6  should  however 
not  be  exceeded,  unless  there  be  some  other  important 
reason  for  a  higher  ratio. 

151.  Of  calculating  the  Numbers  for  Wheel-Work. 
Let  n  be  the  number  of  revolutions  per  minute  for  the  first 
axis^  to  which  the  moving  power  gives  motion ;  and  n  the 
number  of  revolutions  per  minute  of  the  last  axis,  where 

the  resistance  or  working  point  is.     Then,  n  :  n  : :  1  :  — » 

which  is  the  ratio  the  velocity  is  to  be  increased  or  di- 
mimshed. 

If  this  ratio  shpuld  not  exceed  1:6a  single  wheel  and 


118  ON   THE   TEETH    OF   WHEELS.  [CHAP.  IV^* 

pinion  will  be  sufficient ;  but  when  it  exceeds  that  ratioy 
more  will  be  necessary. 

When  each  of  the  pinions  has  the  same  number  of  teedx, 
and  each  wheel  the  same  number ;  then,  the  ratio  of  th.^ 
number  on  a  pinion,  will  be  to  the  number  on  a  wheel 

as  1   :  ^;  oras  1  :  (^V'  .,   ^ 

The  number  of  pinions  being  a. 

But  it  is  sometimes  necessary  to  adapt  the  trains  of  ma- 
chinery to  produce  diflferent  velocities  at  the  working  points, 
and  it  is  on  this  account  often  desirable  to  vary  the  size  of 

the  wheels.  Then,  the  ratio  1  :  -  must  be  decomposed  into 

factors  suitable  to  the  nature  of  the  work  to  be  done ;  if 
those  factors  be  any  numbers  a,  b,  c,  &c.  the  ratio  will  be 

1   :  a  X  6  X  c,  &c.  :  —  (2.) 

The  first  mover  should  be  as  near  as  possible  to  the  re- 
sistance. But,  when  it  is  absolutely  necessary  to  perform 
operations  at  a  considerable  distance  from  the  first  mover, 
the  velocity  of  the  communicating  shafts  should  be  brought 
up,  as  near  as  possible,  to  the  first  mover,  to  that  which  is 
most  advantageous  for  the  difierent  species  of  work. 

But  the  velocities  of  the  parts  of  machines  are  often 
given  in  feet  per  second ;  let  v  be  the  velocity  in  feet  per 
second ;  then  60 1;  is  the  feet  described  in  a  minute.  Also 
let  d  X  3*1416  be  the  circumference  which  moves  with  the 

velocity  V ;   then  g,^^^^^  =  —^ —  being  the  revoluticms 

which  the  axis  makes  in  a  minute. 

When  the  first  mover  acts  with  a  velocity  v  at  the  dis- 

,    .  -.         ,         .       ^      10-09t7 
tance  ^  a  from  the  axis;  then  — j —  =  n ;  and  the  ratio 


CIMP.  IV.]  ON   THE    TEETH    OF    WHEELS.  119 

6f  the  teeth  on  tlie  pinions  should  be  to  those  on  the  wheels 

(3.) 


^9f)-09tt 
And,  when  the  velocity  v  of  the  working  point  is  also  given, 


19-09  V 


■  N ;    and    the   ratio  of  the    teeth   on    the 


nitiionsshouldbeto  those  on  the  wheels  as  1  :  ( — \z        (4.) 

If  the  velocity  is  to  be  decreased,  then  it  will  be  as  the 
fheeU  are  to  the  pinions,  instead  of  the  pinions  to  the 
wheels.  The  use  of  these  proportions  will  be  best  illus- 
irated  by  examples. 

I5'i  Example  I.  Let  it  be  required  to  calculate  the 
numbers  for  a  com  mill  moved  by  an  overshot  water-wheel. 
This  case  comes  under  Proportion  (l).  where  o,  the 
number  of  pinions,  will  never  exceed  2,  and  not  often  more 
than  1.  And  in  order  that  the  grain  may  not  be  too  much 
heated  in  grinding,  the  velocity  of  the  circumference  of  the 
miUstonc,  should  not  he  greater  than  23  feet  per  second, 
hence  v  =  23  ;  and  d  will  be  the  diameter  of  the  millstone 
in  feet. 

153.  If  we  suppose  the  wheel  and  its  concomitants  to 
offer  no  resistance  to  the  impulse  of  the  water,  it  is  manifest 
that  the  velocity  of  the  circumference  will  then  be  the 
wme  as  that  with  which  the  water  strikes  it,  or  that  which 
.ue  to  the  whole  height  of  the  fall,  and  will  therefore  be 
iS9ed  by  V  =  -J  '■Zgh;  where  h  denotes  the  height  of 
fall  in  feet ;  «  the  velocity  in  feet  per  second,  and  g  = 
feet,  the  velocity  generated  by  gravity. 
But  it  is  not  consistent  with  the  laws  of  nature  that  a 
machine  can  he  put  in  motion  without  offering  some  re- 
liance to  the  moving  power ;  for,  in  the  first  place,  the 


120  ON   THE   TEETH    OF   WHEELS.  [CHAP.  IV. 

friction  of  the  parts  has  to  be  oyercome,  and  it  must  be  a 
machine  of  a  very  simple  construction  indeed,  if  the  ac- 
complishment of  this  alone  does  not  expend  one-half  of  the 
force  applied :  but  in  the  generality  of  combinations,  it 
will  be  found  to  balance  nearly  two-thirds  of  the  power  ap- 
plied ;  we  will  therefore  be  pretty  near  the  truth  by  as- 
suming the  friction  as  equivalent  to  two-thirds  of  the 
moving  power. 

In  the  next  place,  there  is  the  resistance  to  be  overcome 
at  the  working  point,  and  this  is  equivalent  to  the  quan- 
tity of  work  to  be  performed,  which  must  therefore  expend 
the  remaining  third  of  the  moving  power.  ^  It  thence  ap- 
pears, that,  in  the  case  of  an  overshot  water-wheel,  the 
height  of  the  fall  which  produces  the  velocity  of  the  cir- 
cumference, must  be  divided  into  two  parts ;  one  part  to  over- 
come the  friction,  and  the  otiier  to  produce  the  useful 
effect 

Let  e  be  that  part  of  the  fall  which  is  competent  to  over- 
come the  friction  of  the  loaded  machine  only;  or  that 
which  corresponds  to  the  velocity  when  the  useful  effect  is 
nothing ;  and  let  x  be  that  part  of  the  fall  producing  the 
velocity  corresponding  to  the  maximum  of  useful  effect 
It  is  therefore  manifest,  that  the  effective  force  of  the  water 
on  the  wheel,  when  the  work  done  is  the  greatest  possible, 
will  always  be  proportional  to  A  —  a: ;  and  when  the  work 
done  is  nothings  the  effective  force  will  always  be  proper- 
tional  to  h  —  e.  Now,  the  difference  of  these  two  quan- 
tities drawn  into  the  velocity  must  be  a  maximum  when 
the  greatest  effect  is  obtained ;  hence  we  have  v  (A  —  or  — 
A  -h  ^)  =  ?^  (^  —  a:)  a  maximum.  But,  by  the  laws  of  fall- 
ing bodies,  we  have  v=  >/  Q,gx\  let  this  be  substituted  for 

V  in  the  above  expression,  and  it  becomes  ^/  Q,gx  (e  —  x)y 
a  maximum ;  or  by  dropping  the  constant  factor  2  g*, 
it  is  e  X   ^  or  —  dL  maximum.      Let  this  expression  be 


CHAP.IY.J  ON   THE   TEETH    OF   WHEELS.  121 

thrown  into  fluxions  and  equated  with  zero  or  nothing, 
and  it  becomes 

^  €  <jb         X  ■""  <u  «ir    X  —  v/» 

Gmsequentlj,  by  transposition  and  reduction,  we  get 

Bat  we  have  stated  above  that  two-thirds  of  the  moving 
power  is  expended  in  overcoming  the  friction ;  hence  we 
ha?e«  =  A  — f-A  =  JA;  therefore,  ^=iA,  and  the  velo- 
city corresponding  to  or  is 


t?=  >/64ixjA  =  2-673  ^h. 

We  have  next  to  determine  the  diameter  of  the  wheel 
in  relation  to  the  heigl;it  of  the  fall  when  the  effect  is  a 
maximum ;  and  for  this  purpose,  let  ^  be  equal  to  that  por- 
tion of  the  circumference  which  is  loaded  with  water,  and  x 
equal  the  arc  comprehended  between  the  point  of  impact 
and  the  horizontal  radius ;  then,  by  the  principles  of  men- 
suration, the  magnitude  of  the  solid  which  represents  the 

effective  force,  is  ^  i  {jLH V  where  h  is  the  section  of 

the  stream  supplying  the  buckets.  But  this,  by  the  ques- 
tion, is  to  be  a  maximum.  Let  it  therefore  be  thrown  into 
fluxions,  and  put  equal  to  zero,  and  we  get 

from  which  by  transposition  we  get 

and  reducing  the  quadratic  by  the  rules  of  algebra,  it  be- 
comes 

■ar=^(l-^/f). 
But  1  -  >/i= 1  - 70711  =-29289 ;  hence  we  get 


ar=-29289^;  and^  = 


•29289 


122  ON  THE  TEETH   OF  WHEELS.  [CHAP.  IV. 

It  is,  however,  obvious,  that  the  difference  between  ihe 
two  arcs  <f>  and  a:,  must  be  equal  to  a  quadrant ;  that  is, 

,gQggQ-ar=90^  and  this  gives  x^SJ""  1&. 

Let  r  denote  the  radius  of  the  wheel,  estimated  from 
the  centre  to  the  remote  point  of  the  bucket ;  then  will 
r  (1  4- sin.  37*  16')  express  the  eflGective  height  of  the  ML 
But  the  natural  sine  of  37**  16',  is  '60553 ;  hence  we  have 
1*60553  r  for  the  effective  height;  and  if  the  absolute 
height  of  the  fall  be  equivalent  to  nine-eighths  of  the 
effective  height,  we  shall  have  A=l*8062  r ;  consequently, 
r  —  *554f  h  and  rf=2  r=l*108  h.  Having  thus  determined 
the  values  of  v  and  d^  let  them  be  substituted  in  the  ap- 
propriate Proportion  number  (4)  preceding,  and  we  get 

1  :  ^ — ^ — ,  when  there  is  only  one  pinion  j 

but  when  there  are  two  pinions,  it  becomes 

asl:{?:^}*. 

In  practice,  the  height  of  the  fall  and  the  diameter  of 
the  millstone  will  always  be  known ;  in  the  present  in- 
stance let  D  =  5  feet,  and  A  =  16   feet ;  then  we  have 

>/  16  =  4,  and  therefore  it  is 

^  .  9*54  X  4 

X     ■  f 

5 
or  as  1  :  7*632.  Now  it  will  be  better,  in  this  case,  to  have 
two  pinions,  since  the  ratio  is  greater  than  1:6;  there- 
fore as  1  :  7*632^  or  as  1  :  2*763,  so  is  the  teeth  in  each 
pinion  to  the  teeth  in  its  wheeL  And  making  the  prime 
number  11  the  number  of  teeth  of  each  pinion,  we  shall 
have  1 1  X  2*763  =  30,  the  nearest  whole  number  for  the 
teeth  of  each  wheel.  The  thickness  of  the  teeth  being 
found  by  the  rules  for  that  purpose,  (see  Art  137,)  the 


CHAP.  IV.]  ON    THE   TEETH    OF    WHEELS.  I'iS 

radius  of  the  wheels  and  pinions  will  be  found  by  the  table 
of  radii,  (Art.  14.5.) 

;ijain,  let  the  fall  be  4'8-l-  feet,  the  diameter  of  the  mill- 
stone being  5  feet  as  before ;    then,    .j  k  =    'i'Q,    and 

1: —  becomes  as  1  :  4'1976.    Here  one  pinion  or  trun- 

D 

file  will  be  best ;  and  making  the  teeth  on  the  pinion  1 1 
those  on  the  wheel  will  be  11  x  4'-1976  =  4i6  in  the  nearest 
wliolc  number. 

The  young  millwright  will  find  it  useful  to  calculate  a 
table  by  these  rules,  which  might  be  extended  to  under- 
shot wheels,  and  exhibit  at  one  view  the  whole  construction 
mills. 

154-.  Example  II.  Let  it  be  required  to  arrange  the 
lumbers  for  a  machine  for  raising  water,  where  the  pumps 
are  to  make  N  strokes  per  minute,  the  velocity  of  the 
moving  power  being  v  feet  per  second,  and  the  diameter  of 
the  circle  described  by  the  power  d  feet. 

This  case  is  an  example  of  the  use  of  Proportion  (3),  or 

1 :  (^ — 1„ — J-.    Now  let  the  moving  power  be  a  horse, 

where  v  is  2^  feet  per  second,  (Art.  119,)  and  d  the  dia- 
meter of  his  track,  30  feet ;  the  pump  to  make  20  strokes 


em 

r 


minute,  or  n=20,  then  the  ratio  is  I  : ■.,  t 

19-09  x^i^ 


1  :  1'2*6  nearly.     Therefore  if  the  pinion  have  13  teeth, 
wheel  should  have  13x  12-6  =  164  teeth  in  the  nearest 
whole  numbers. 

But  we  should  prefer  making  two  piniona,  and  then  the 

ratio  will  be  1  :  v  liJ*6  or  as  1  :  3-549  nearly,   and  each 

pinion   having    1 1    teeth,   the  wheels   should    each    have 

H]  X  3-549  =  39  teeth. 

^HfThese  examples  will  perhaps  be  sufficient  to  explain  the 
^^bdc  of  calculation ;  and  when  once  it  is  understood,  it 
^^Bl  vur)-  easily  be  applied  to  other  cases ;   and  will  be 


134  ON   THE   TEETH   OF   WHEELS.  [CHAP.  IV. 

found  somewhat  more  convenient  than  the  methods  usually 
followed. 


PRACTICAL   OBSERVATIONS   WITH   REGARD   TO   THE   MAKING   OF   PATTERNS 

FOR   CAST   IRON    WHEELS. 

155.  Having  determined  the  pitch  of  the  wheel  strong 
enough  for  the  purpose  to  which  it  is  to  be  applied,  the 
thickness  of  the  tooth  serves  to  regulate  the  proportionate 
strength  of  the  other  parts. 

A  very  respectable  millwright  informs  me,  that  he  has 
for  a  considerable  time  adopted  the  following  rule  for  de- 
termining the  length  of  the  teeth  of  wheels,  the  practical 
efficacy  of  which  he  has  found  quite  satisfactory. 

Rule. — Make  the  length  of  the  teeth  equal  to  the  pitchy 
deducting  freedom^  (by  the  freedom  is  meant  the  distance 
at  the  top  of  one  tooth,  and  the  root  of  another  measured 
at  the  line  of  centres,)  in  other  words,  the  distance  from 
root  to  root  of  the  teeth,  at  the  line  of  teeth  when  the 
wheels  are  in  action,  exactly  equal  to  the  pitch. 

For  example — ^he  makes  the  teeth  of  two  inches'  pitch,  1 
inch  and  ii  in  length,  which  is  allowing  iV  of  freedom. 

Another  respectable  millwright,  who  has  had  much  ex. 
perience,  particularly  in  miUs  moved  by  horses,  has  for  a 
considerable  time  past  made  the  teeth  of  his  wheels  in 
length  only  one  half  of  the  pitch,  and  works  them  as  deef 
as  possible  without  the  point  touching  the  bottoms.  Be- 
fore he  fell  on  this  expedient,  he  found  the  teeth  exceed- 
ingly  liable  to  be  broken  fix)m  any  sudden  motion  of  the 
horses*. 

Indeed,  ujwn  reflection,  it  will  be  found  there  is  no  oc- 
casion for  more  freedom,  than  that  the  point  of  the  tooth 
of  the  one  wheel  shall  just  clear  the  ring  of  the  other ; 

*  lUcspccting  the  length  of  icclb,  sscc  Art.  2D,  42,  luid  47. 


MAP.  IV,]  ON    THE    TEETH    OF   WHEELS.  195 

more  than  this  must  only  serve  to  weaken  the  teeth.  The 
Diixle  of  gearing,  however,  ahove  alhided  to,  is  more  neces- 
sary in  horse  mills  than  where  the  moving  power  is  steady 
lad  regular. 

HattoQ  (on  Clock-work)  recommends  making  the  dis- 
lance  of  the  pitch  line  |  of  what  we  call  the  thickness  of 
the  tooth.  Thus,  suppose  the  rule  applied  to  a  two  inch 
pitch  and  that  the  tooth  and  space  were  exactly  equal,  then 
the  tooth  would  project  f  of  an  inch  beyond  the  pitch  line, 
(nil  its  root  would  be  as  far  within  the  pitch  line,  as  to  re- 
eeive  freely  the  tooth  intended  to  act  on  it:  suppose  it  also 
J,  then  the  tooth  would  be  1 J  inch  long,  besides  the  free- 
dom, which,  making  as  above,  iV,  the  tooth  would  be  in 
»11  lU  inch  long. 

156.  But  it  is  to  be  remarked,  that  the  millwright,  in 
naking  his  pattern  for  a  cast  iron  wheel,  has  to  attend  to 
I  drcumstance  arising  from  the  nature  of  that  material. 
The  pattern  must  not  only  be  of  such  a  form  as  to  be  suf- 
ficiently strong,  calculating  by  the  bulk  of  the  parts,  but 
|lso  proportioned,  so  that  when  the  fluid  metal  is  poured 
B  the  mould,  it  may  cool  in  every  part  nearly  at  the  same 
bie. 
HTien  due  attention  is  not  paid  to  this  circumstance,  as 
e  metal  is  cooling,  if  it  contract  faster  in  one  part  than 
I  another,  it  will  be  apt  to  break  somewhere,  just  as  a 
haking  glass  is  broken  by  suddenly  cooling  or  heating  in 
ly  particular  part  of  it.  In  all  patterns  for  cast  iron, 
kout  J  of  an  inch  to  the  foot,  should  he  allowed  for  the 
Btraction  of  the  metal  in  cooling. 

Attention  must  also  be  paid  to  taper  the  several  parts, 
that  they  may  rise  freely  without  injuring  the  mould, 

1  the  founder  is  drawing  them  out  of  the  sand.     A 
lie  observation  of  the  operations  of  a  common  foundry, 
1  better  instruct  on  this  part  of  the  subject  than  many 
We  niav  obser\'C,  however,   that  about  Vs   of  an 


126 


ON  THE  TEETH   OF   WHEELS. 


[chap. 


inch,  in  a  depth  of  6  inches,  is  commonly  a 
taper. 

157*  Attending  to  those  circumstances,  weoffer  the  folio 
ing  proportions  as  having  heen  found  to  answer  in  practi 

Make  the  thickness  of  the  ring  a  e  equal  to  the  thie 
ness  of  the  tooth  a  c  near  its  root.     When  the  ring* 
made  thinner  than  the  root  of  the  tooth,  the  ring  common, 
gives  way  to  a  strain,  which  would  not  hreak  the  tooth. 

Make  the  arm,  at  the  part  where  it  proceeds  from 
ring,  of  the  same  hreadth  and  thickness  as  the  ring ; 


18 

'7 


Pio.  2. 


Fig.  3. 


Fio.  4. 


at  the  junction  «» ^  let  it  he  so  formed  as  to  take  off  any 
acute  angle  which  would  he  apt  to  break  off  in  sand. 


CHAP.   IV.2  ^^   THE   TEETH    OF   WHEELS.  127 

The  arms  should  become  larger  as  they  approach  the 
centre  of  the  wheel,  (see  Emerson,  Prop.  119,  Rule  8,) 
and  the  eye  e,  should  be  sufficiently  strong  to  resist  the 
driving  of  the  wedges,  by  means  of  which  it  is  to  be  fixed 
on  the  shaft.     This  cannot  be  brought  easily  to  calculation. 

On  the  other  hand,  care  must  be  taken  not  to  make  the 
eye  so  thick  as  to  endanger  unequal  cooling. 

It  should  be  somewhat  broader  than  the  breadth  of  the 
teeth,  in  order  that  it  may  be  the  firmer  on  the  shaft :  this 
breadth  must  be  greater  in  proportion  as  the  wheel  is  large. 

When  the  ring  a  e  is  about  an  inch  thick,  it  is  common 
to  make  the  eye  about  an  inch  and  a  quarter  thickness,  and 
about  one-fifth  broader  than  the  ring,  when  the  wheel  is 
about  four  feet  diameter. 

Small  wheels  have  generally  but  four  arms,  but  it  being 
improper  to  have  a  great  space  of  the  ring  unsupported,  the 
number  of  arms  should  be  increased  in  large  wheels. 

In  order  to  strengthen  the  arms  with  little  increase  of 
metal,  it  is  not  unusual  to  make  them  feathered,  which  is 
done  by  adding  a  thin  plate  to  the  metal  at  right  angles 
to  the  arm,  as  represented  by  figure  third.  Fig.  4  is  a 
section  of  Fig.  3,  at  a  b. 

The  same  rules  apply  to  bevelled  wheels ;  of  the  prac- 
tical mode  of  laying  down  the  working  drawings  of  which 
we  have  already  spoken.  But  it  is  proper  to  observe,  that 
the  eye  of  a  bevelled  wheel  should  be  placed  more  on  that 
side  which  is  furthest  from  the  centre  of  the  ideal  cone  of 
which  the  wheel  forms  a  part. 

158.  When  wheels  are  beyond  a  certain  size,  it  becomes 
necessary  to  have  patterns  sometimes  made  for  them,  cast 
in  parts,  which  are  afterwards  united  by  means  of  bolts. 

To  prevent  the  bad  eflects  of  unequal  contraction,  the 
arms  may  be  forked  or  curved,  as  in  the  second  figure ;  the 
forked  or  curved  parts  are  commonly  of  the  same  radius 
as  tihe  wheel,  and  spring  from  the  half  length  of  the  arms. 


128  ON   THE   TEETH    OF   WHEELS.  [CHAP.  !>?"• 


MATBBIALS  OF  PATTERNS. 


1 59.  The  patterns  should  be  made  of  well-seasoned  wooci 
The  most  proper  is  clean  mahogany*,  but  that  being  no^^ 
very  expensive,  white  deal  is  most  commonly  used.  BeeaXl 
is  very  often  used  for  the  teeth,  and  being  a  close  graine*^ 
wood,  it  may  be  made  very  smooth. 

It  is  almost  superfluous  to  say,  that  the  workmanship  of 
wheel  patterns  should  be  such  as  to  produce  great  accuracy^ 
and  a  smooth  surface,  the  former  being  essential  to  the 
good  movement  of  the  wheels,  and  the  latter  to  make  the 
patterns  produce  a  good  and  clean  impression  in  the  sand 

It  is  a  conmion  practice,  in  many  places,  to  make  teeth 
very  large  in  the  pattern,  and  after  fixing  the  wheels  on 
their  shafts,  to  chip  and  file  the  teeth  to  the  proper  size ; 
but  we  doubt  whether  this  practice  be  really  advantageous ; 
for  besides  the  great  time  which  it  occupies  thus  to  dresa 
the  iron  teeth,  and  the  consequent  expense,  there  is  the 
loss  of  the  outer  skin  (if  we  may  use  the  expression)  of  the 
cast  iron,  which  is  by  far  its  most  smooth  and  durable  part. 
In  cotton  mills,  therefore,  this  absurd  method  is  now  bat 
seldom  practised  t. 

*  The  common  chestnut  tree  is  equal  to  any  wood  that  can  bo  used ;  and 
its  dimensions  adapt  it  equally  well  for  moulds  with  Honduras  mahogany. 

t  Messrs.  Peel,  Williams,  and  Co.,  have,  after  great  time,  trouble,  and 
expense,  made  and  arranged  a  very  great  number  of  patterns  of  wheels,  so 
as  to  suit  almost  every  case  that  can  in  practice  occur.  They  have  published 
a  complete  list  of  them,  which  they  intend  inserting  also  in  the  '^  Repertory 
of  Arts."  In  my  opinion,  what  they  have  done  is  a  material  national  be- 
nefit ;  their  expense,  I  am  informed,  for  patterns,  has  not  been  less  than 
four  thousand  pounds.  There  is,  however,  every  reason  to  think,  that  it 
will  be  an  excellent  thing  ultimately  for  themselves,  as  well  as  of  great 
practical  utility  to  the  public. — Buchanan. 


CHAPTER  V. 


»  THE  USE  OP  CHARTS,  AND  SOME  FURTHER  EXPLANATION 
[  0?  THE  CONSTRUCTION  OF  THE  TABLES  OF  PITCHES  OF 
[  VHEEL-WORK. 

.  When  quantities  of  any  kind,  such  as  time,  space, 
My,  &c.,  expressed  in  numbers,  are  mentioned,  it  often 
s  a  painful  exertion  of  the  mind  to  recollect  and 
e  them.  Hence  the  utility  of  bringing  them  into  one 
r  in  tables.  But  there  is  another  motlc  of  comparing 
iDtities  not  so  generally  practised,  though,  in  many 
I  much  more  easy  and  satisfactory  to  the  mind.  I 
I  to  charts,  in  which,  instead  of  using  figures,  as  in 
tafiles,  the  quantities  are  geometrically  represented.  This 
■  done  by  dinding  the  sides  of  a  square  or  rectangle  into 
'  i|ual  parts,  and  drawing  parallel  lines  at  right  angles  from 
tiL' divisions.  The  quantities  are  pointed  off  at  certain 
itersections  of  these  scales. 

llli.  WTicn  the  quantities  increase  or  decrease  in  arith- 
in'tical  proportion,  as  1,  2,  3,  4,  &c.,  that  proportion  will 
!k'  represented  by  a  straight  line,  which  will  pass  through 
thfse  points. 
\(>'2.  But  supposing  the  quantities  to  increase  in  geome- 
ial  proportion,  as  1,  4,  9,   l6,  &c.,  the  line  passing 
'rdugh  the  points  of  intersection  will  form  a  curve. 
These  two  cases  will  be  best  explained  by  examples. 
K).'?.  F^rst,  Suppose  the  value  of  any  thing  to  increase  as 
'eight, — the  scale  on  the  one  side  of  the  square  will 
I  represent  the  value,  and  that  on  another  the  weight. 
at  AC,  I'ig.  I.  Plate  I.,  represent  weight,  (say  ounces.) 
AD  value,  (say  shillings.)     Now,  suppose  we  mark  tho 


130  ON    THE   TEETH    OF   WHEELS.  [CHAP.  Y. 

price  of  four  ounces,  it  is  done  by  placing  a  dot  opposite  to 
four,  on  the  line  of  value,  and  opposite  to  four  on  the  line 
of  weight,  at  the  intersections  of  the  perpendiculars  from 
these  points,  which  intersection  is  marked  by  d  on  the  figuie. 
In  the  same  manner,  we  may  mark  the  value  of  1  2,  3, 5, 
6  ounces.  These  points  are  marked  a  b  c  e  fy  and  the 
straight  line  a  b  passes  through  them  all,  and  shews  the 
regular  progress  of  the  proportion. 

It  is  of  no  consequence  whether  the  divisions  on  the  line 
A  c  be  greater  or  less  than  those  on  a  d,  provided  the  lines 
be  divided  into  equal  parts. 

164.  Second y  Suppose  an  accelerating  motion,  such  as 
that  of  a  falling  body,  is  to  be  laid  down  on  a  chart, — ^this 
motion  increases  as  the  squares  of  the  times ;  that  is,  the 
body  falls  a  certain  distance  in  the  first  second  of  time,  four 
times  that  distance  in  the  next  second,  and  nine  times  in 
the  third  second,  &c. 

These  points  are  accordingly  marked  in  Figure  ^  by  a 
opposite  to  1  on  both  scales,  by  b  opposite  2  on  the  scale 
of  time,  A  D  and  4  on  that  of  motion  a  c  ;  by  c,  opposite  S, 
on  a  d,  and  9  on  a  c,  &c.,  the  line  a  b  passing  through  these 
points  forms  a  curve. 

165.  When  the  proportion  of  any  kind  is  regular,  the 
curve  has  a  regular  easy  sweep ;  if  otherwise,  the  curve  will 
undulate,  or  have  irregular  windings.  Hence,  it  is  a  good 
mode  of  proving  many  kinds  of  tables,  to  lay  down  the 
quantities  thus  geometrically ;  for  if  there  be  any  material 
error,  when  the  proportion  ought  to  be  regular,  an  elbow 
will  appear  in  the  line  a  b. 

166.  Much  calculation,  too,  may  often  be  saved,  for  when 
a  few  of  the  principal  points  at  some  distance  from  each 
other  are  obtained  in  the  curve,  the  rest  of  it  may  be  easily 
found,  by  drawing  the  curve  between  them  with  a  slip  of 
thin  wood,  or  any  other  such  means  of  producing  an  easy 
curve. 


CHAP.   V.3  ON    THE   TEETH   OF   WHEELS. 


131 


167.  Charts,  on  similar  principles,  are  used  for  many 
purposes ;  for  example,  there  are  biographical  charts,  show- 
ing the  periods  when  eminent  men  appeared,  and  the  rela- 
tive length  of  their  lives.  They  are  also  used  for  represent- 
ing revenue  of  any  kind,  which  generally  forms  an  undu- 
lating line,  as  does  also  the  charts  of  the  heights  of  the  ba- 
rometer, or  the  temperature  indicated  by  the  thermometer. 
The  heights  of  mountains,  the  tides — ^in  short,  they  may  be 
considered  as  merely  scales  of  equal  parts,  and,  of  course, 
are  applicable  to  all  subjects  capable  of  being  represented 
by  numbers. 

168.  In  order  to  give  a  more  distinct  comparative  view 
of  the  tables  of  pitches,  in  the  ^^  Essay  on  the  Teeth  of 
Wheels,*'  we  shall  lay  their  contents  down  in  one  chart,  but 
previously  collect  all  these  tables,  and  give  some  further 
explanation  of  their  mode  of  construction. 


TABLES  OP  PITCHES  OF  WHEEL-WORK. 
(See  Chap.  IV.,  Art.  123—129.) 

TABLE  I*. 

Velocity  of  the  pitch  line  being  3  feet  per  second,  and 
breadth  of  teeth  9  inches. 


w 

X 

Y 

Z 

Htehin 
inches. 

Breedth 

of  teeth  in 

inches. 

Value  of 

strength 

in  hontcs* 

power. 

Value  of 

strength 

in  honics' 

power. 

4 

8i 
3 

2i 
2 

1 

9 
9 
9 
9 
9 
9 
9 

16- 
12-25 

9- 

6-25 

4- 

2-25 

1- 

12- 
10-5 

9- 

7-5 

6- 

4-5 

3- 

*  Tables  I.  and  11.  are  for  teeth  attached  to  water-wheels,  where  liable 
to  be  worn  by  aand  and  water. 

k2 


las 


ON  THE  TEETH  OF  WHEELS. 


[CH 


TABLl  U. 


The  velocity  being  3  feet  per  second,  and  the  bi 
of  the  teeth  double  each  pitch. 


w 

X 

Y 

Z 

Ktcfain 

Twice  the 
Dieedth. 

Value  of 

Itrength 

in  hones' 

power. 

Value  of 

ttrmgth 

in  homt* 

power. 

4 

3* 
3 

2 

1* 

1 

8 
7 
6 
5 
4 
3 
2 

14-22 
9-53 
6- 

3-47 

1-77 

•75 

•22 

10-66 
8-17 
6- 

4-16 
2-65 
1-5 
'66 

TABLB  in*. 


The  velocity  being  1 1  feet  per  second,  and  the  br 
of  teeth  8  inches. 


w 

X 

Y 

Z 

Pitch  in 
inches. 

BreMlth 

of  teeth  in 

inches. 

Value  of 

strength 

in  horses* 

power. 

Value  of 

strength 

in  horses* 

power. 

4 

3i 
3 

H 

2 
1 

00  00  00  00  00  00  00 

81-77 
62-61 
46- 
31*94 
20-44 
11-5 
5*11 

61-33 

53-66 

46- 

38-33 

30-66 

23* 

15-33 

•  TiWce  III^  IV^  V^  and  VI.  are  for  teeth  properly  greased, 
from  sand. 


UP.  v.] 


ON   THE   TEETH    OF   WHEELS. 


133 


TABLE    IV. 


The  velocity  being  1 1  feet  per  second,  and  the  breadth 
)able  each  pitch. 


w 

X 

Y 

Z 

Pitrhin 
inches. 

Twice  the 
pitch  in 
breadth. 

Value  of 

strength 

in  hones* 

power. 

Value.of 

strength 

in  hones* 

power. 

4 

3i 

3 

H 
2 

H 
1 

8 
7 
6 
5 
4 
3 
2 

81-77 
54-78 
34-5 
19-95 
10*22 
4-31 
1-28 

61-33 
46-95 
34-5 
23-94 
15-33 
8-62 
3-84 

TABLE  V. 


The  velocity  being  3  feet  per  second,  and  breadth  of 
Beth  8  inches. 


w 

X 

Y 

Z 

Pitch  in 
inches. 

Breadth 

of  teeth  in 

inches. 

Value  of 

strength 

in  horses* 

power. 

Value  of 

strength 

in  horses' 

power. 

4 

3 

H 
2 

1 

8 
8 
8 
8 
8 
8 
8 

22-30 
1707 
12-54 

8-71 
5-57 
3-13 
1-39 

16-72 

14-63 

12-54 

10-45 

8-36 

6-26 

417 

134 


ON   THE   TEETH   OF  WHEELS.  [CHAP.  Y. 


TABLB  Vr. 


The  velocity  being  3  feet  per  second,  and  breadth  double 
each  pitch. 


w 

X 

Y 

Z 

PHchin 
inches. 

Twice  the 
pitch  in 
breadth* 

Value  of 

strength 

in  hones' 

power* 

Value  of 

strengui 

in  hones' 

power. 

4 

3 

H 
2 

1 

8 
7 
6 
5 
4 
3 
2 

22-30 
14-93 
9-4 
5*44 
2-78 
117 
0-34 

16-72 
12-79 
9-4 
6-53 
4-17 
2-34 
1-02 

BEFEBENCB  TO  TABLB  I*,    ART*   123,   IN  THB  E8SAT  ON  XHB 

TBBTH  OF  WHBBL8* 

169.  Column  X  is  omitted  in  Tables  I.,  III.,  and  V., 
being  only  a  repetition  of  the  same  breadth  for  all  the 
pitches  of  each  table,  but  as  being  perhaps  plainer,  they 
are  inserted  here. 

The  numbers  in  column  y  are  found  by  squaring  the 
pitch  in  column  w. — (See  Proposition  I.  p.  83.) 


EXAMPLE. 

The  square  of  4,  (the  pitch  in  inches)  >=  16,  the  value 
of  strength  in  horses*  power. — (See  the  first  line  of  table.) 

The  column  z  is  found  by  inverse  proportion. — (See 
Prop.  II.  p.  85.)  taking  three  inches,  (the  standard  pitch,) 
always  as  the  first  term,  the  pitch  column,  w,  as  the 
second,  and  the  horses'  power,  in  column  y,  as  the  third 
term. 


CHAP,  v.]  ON   THE   TEETH   OF   WHEELS.  135 


EXAMPLE. 

hi      In.  HonaT  power.  Honw*  power. 

3:4::  16  :  12  (See  first  line  of  table.) 
3:1::     1  :     3  (See  last  line  of  table.) 


&BPBRSNCB  TO   TABLE  II. 


170.  The  numbers  in  column  y  are  found  here  by  direct 
proportion  from  Table  I.,  nine  inches  (the  breadth  in 
TaMe  I.)  being  always  the  first  term  of  the  proportion ; 
the  horses'  power  in  y,  Table  I.  the  second,  and  the 
breadth  in  x.  Table  II.  the  third  term. 


EXAMPLE. 

iB-Bomr  power.     In.     R 


9 :  16:  :  8  :  14-22  (See  Table  II.  line  first).  Column 
2  is  found,  as  in  Table  I.  by  inverse  proportion,  3  inches 
(the  standard  pitch)  being  always  the  first  term.     Thus, 

^     Ib.  Honei^  power.  Honei*  power. 

3:4::  14-22  :  10-66.— (See  first  line  of  table.) 


RBPBRBNCB   TO   TABLB   III. 


171.  The  numbers  in  column  y  are  found  by  direct  pro- 
portion, taking  9  (the  square  of  the  standard  pitch  of 
three  inches)  as  the  first  term,  and  the  square  of  the  pitch 
in  w  as  the  second  term. 


EXAMPLE. 

As  9  (the  square  of  3  inch  pitch) 

Is  to  16  (the  square  of  4  inch  pitch). 

So  is  46  horses'  power  (the  value  in  column  y  of  3  inch 

pitch) 
To  81*77.— (See  first  line  of  table.) 


136  ON  THE  TEETH  OF  WHEELS.      [CHAP.  Y. 

Column  z  is  found  by  inverse  proportion,  as  in  former 
tables. 


EXAMPLE. 

In.        In.  Honcft*  power.  Hones*  power. 

3  :  4  :  :  8177  :  61-33 (See  Istlme  of  table.) 


BEFKBBNCB  TO  TABLE  IV. 


172.  The  numbers  in  column  y  are  found  here  in  a 
manner  similar  to  Table  II.  by  direct  proportion. 


EXAMPLE. 

In.        In.  Hortc**  power.    Horaet*  power. 

8:7::  62-61  :  54-78.— (See  2d  line  of  table.) 
The  numbers  in  column  z  are  found,  as  in  the  former 
tables,  by  inverse  proportion, 

In.        In.  Hortct*  power.    Hones*  power. 

3  :  4  :  :  8177  :  61-33.— (See  1st  line  of  table.) 


KEFEBENCE  TO  TABLE  V. 


173.  The  numbers  in  column  y  of  this  table  are  found 
by  direct  proportion  from  column  y  of  Table  IV.,  1 1  feet 
(velocity  per  second)  being  always  the  first  term,  and  3 
feet  (velocity)  the  second  term. 

Ft.  Ft.  Hones' power.   Horaes' power 

Thus,  11  :  3  :  :  81-77  :  22-30.— (See  Istlme  of  table.) 
Column  z  is  found  by  inverse  proportion,  as  in  all  the 
former  tables : 

In.        In.        Horses'  power.  Horses*  power. 

Thus,  3:4::  22-30  :  16'72.— (See  Istlme  of  table.) 


RBFBBBNCB   TO   TABLB  VI. 

174.  The  numbers  in  column  y  and  z  in  this  taUe^  are 


I: CHAP,  v.]  ON    THE    TEETH    OF   WHEELS,  137 

I  fbnoed  from  Table  V.  in  the  same  mamicr  as  those  columns 
I  in  Table  II,  are  formed  from  Table  I.  The  only  differ- 
lence  in  Table  VI.  from  Table  V.  is  that  which  arises 
■from  the  difference  of  the  breadth  of  the  teeth. 


EXPLANATION  OF  THE  CHABT. 

175.  The  scale  on  the  line  a  b  represents  tlic  pitch  in 
[inches. 

The  scale  on  the  line  a  c  the  horses'  power ;  a  single 

iinple  will  probably  be  sufficient  to  illustrate  the  use  of 

[fbe  chart. 

Suppose  the  pitch  to  be  3^  inches,  let  it  be  required  to 

1  the  horses'  power  to  which  that  pitch  is  equal  when 

Kmg  at  3  feet  per  second,  in  situations  where  properly 

P  greased  and  free  from  sand — observe,  where  the  line  from 

I  file  pitch  3^  intersects  the  curve  z  of  Table  VI.  perpen- 

iitular  to  the  point  of  intersection,  on  the  line  ac,  will  he 

Had  12'79  on  the  scale  or  the  horses'  power  to  which  3^ 

ch,  when  the  teeth  are  7  inches  broad,  is  equal,   after 

Jting  allowance  for  the  length  of  the  teeth. 

OBSERVATIONS. 

176.  Ist,  It  will  he  observed,  that  the  curves  y  and  z 
intersect  each  other,  for  all  the  tables  on  the  pitch  line 
marked  3  inches,  because  that  is  the  standard.  (See  Ist 
Essay,  p.  96,  97-) 

177.  '2d,  The  curves,  continued  from  1  inch  pitch  down- 
irard,  unite  in  the  points  marked  0,  being  the  commence- 
ment of  the  scale  of  pitches,  and  this  part  of  the  curve 

i  the  horses'  power  equal  to  any  fraction  of  an  inch, 

L178.  3d,  It  has  been  already  observed,  p,  98,  that  durability 

Ewell  as  strength  should  be  considered  in  this  investiga- 

Tbe  true  proportion,  therefore,  may  be  somewhere 

len  the  curves  y  and  z,  but  nearer  to  z  than  y.     For 


138  ON  THE  TEETH  OF  WHEELS.      [cHAP.  ▼. 

although  long  teeth  will  be  more  easily  brok^i  than  short 
ones,  yet  while  they  do  not  break,  the  strain  being  gene- 
rally diffused  over  a  greater  number  of  teeth,  they  will 
wear  longer*. 

179.  The  pitch  is  laid  down  on  a  b,  real  measure,  so 
that  if  the  pitch  should  happen  to  be  fractional,  it  may  be 
taken  by  a  pair  of  compasses  and  applied  to  the  chart, 
which  will  at  once  indicate  the  power  to  which  it  may  bet 
equal  at  certain  velocities. 

Among  the  writers  who  have  turned  their  attention  to 
the  forms  of  the  Teeth  of  Wheels,  Professor  Willis,  (^i 
Cambridge,  stands  pre-eminent ;   and  we  are  greatly  io.. 
debted  to  that  gentleman  for  his  liberality  in  permitting 
us  to  insert  the  following  appendix,  his  Essay  on  the  Teeth  of 
Wheels,  and  which  appeared  originally  in  the  second  volume 
of  the  Transactions  of  the  Institution  of  Civil  Engineer^ 
and  for  the  additions  he  has  since  made  to  that  paper. 

•   See  Art.  70. 


APPENDIX    A. 


OH  THE  TEETH  OF  IFHEELS.  BY  R.  WILLIS,  M.A.,  F.H.S., 
H.J1.1NST.C.E.,  JACKSONIAN  PROFESSOll  OF  NATURAL  PHI- 
LOSOPHY   IN    THE   UNIVEESITY    OF   CAMBRIDGE. 

180.  The  investigation  of  the  proper  curves  to  be  given  to 

'k'  teeth  of  wheels,  has  been  a  favourite  occupation  with 

matliematicians  of  the  highest  eminence,  and  the  geometry 

uf  ihe  subject  may  be  considered  to  be  very  nearly  com- 

pk'te. 

Its  application  to  the  requirements  of  modem  construc- 

D  appeared  to  me  to  be  susceptible  of  improvement,  and 

therefore  ventured  to  lay  before  the  Institution  of  Civil 

gineers    the    following    suggestions,    in    which    I    en- 

lToure<l  to  point  out  forms  possessing  properties  more 

I  than  those  hitherto  adopted,  as  well  as  some  prac- 

d  methods  of  tracing  readily  the  outlines  of  the  teeth. 


SECTION   I. 
ON    THE    CURVES    ADAPTED   TO    PRACTICE. 

■There  are  an  infinite  number  of  forms  which  will  answer 

I  conditions  of  enabling  the  teeth  of  one  wheel  to  com< 

plicate  equable  motion  to  those  of  another,  for  it  can  be 

n  that,  under  certain  limitations,  if  any  form  of  tooth 

Segiren,  another  may  be  determined  which  will  work  cor- 


140  ON  THE  TEETH  OF  WHEELS.       [^APPEND.  ^^ 

rectly  with  it*.  A  simple  instrument  which  famishes  a 
practical  solution  of  this  problem,  wiU  probably  carry  mo:«re 
conviction  to  the  minds  of  practical  men  than  the  demtk-^. 
strations  which  have  been  given  by  the  writers  referred  -to 
below.  Let  a  pair  of  boards  he  prepared,  having  th^r 
edges,  AB,  CD,  Fig.  1,  formed  truly  circular.    Attach    to 


one  of  them  by  any  simple  clamp  the  shape  of  the  given 
tooth  E,  cut  out  in  pasteboard,  and  to  the  other  a  piece  of 
stiff  paper  secured  by  means  of  drawing  pins ;  the  shape 
E  must  be  raised  slightly  above  the  surface  of  its  board,  so 
as  to  allow  the  paper  which  is  appended  to  the  other  to 
slide  under  it,  as  is  shewn  in  the  figure.  Make  the  cir- 
cular edges  of  the  two  hoards  roU  together,  and  in  each 
successive  position  draw  the  outline  of  the  shape  e  upon 
the  paper  below  it.  The  result  of  all  these  intersectjng 
lines  will  be  a  bounding  curve,  which  from  the  very  mode 
of  its  description  wiU  touch  the  shape  e  at  some  point  of 
its  edge  in  every  one  of  the  successive  positions.  But  as 
these  positions  were  all  obtained  by  making  one  circular 
edge  roll  upon  the  other,  so  it  is  clear,  that  if  the  new 
curve  be  cut  out  and  made  to  touch  e,  the  motion  produced 
1^  the  mere  contact  of  these  two  curves  wUl  he  exactly  the 

*  Vide  De  la  Hire,  Traite  des  Epicydoidcs.  Young's  Natural  Philo- 
sophy, Vol.  I.  page  176-  Ait;,  Cambridge  Philosophical  TnuiBactioits, 
Vol.  II.  page  277. 


APPEND.  A.]    ON  THE  TEETH  OF  WHEELS.  141 

same  as  that  caused  by  the  rolling  of  the  circular  edges, 
and  therefore  perfectly  uniform. 

Many  forms  of  e,  tried  in  this  manner,  will  prove  un- 
tractable,  for  some  of  the  successive  portions  of  its  edge 
may  cover  up  and  obliterate  parts  of  the  curve  that  have 
been  previously  drawn.  These  are  forms  that  fall  under 
the  limitations  alluded  to,  but  it  is  unnecessary  here  to  in- 
vestigate the  general  reasons  for  this  effect,  as  the  propo- 
sition in  question  is  well  known  and  recognized  by  mathe- 
maticians, although  not  so  well  understood  by  practical 
men. 

From  among  the  infinity  of  curves  that  may  be  offered, 
the  epicycloids  and  involutes  have  been  universally  pre- 
ferred, on  account  of  the  facility  with  which  they  can  be 
mechanically  described,  and  perhaps  because  they  admit  of 
ready  and  independent  demonstrations  of  their  possessing 
the  properties  required.  But  the  practice  has  hitherto 
been  confined  to  that  class  of  epicycloids  which  work  cor- 
rectly with  straight  lines  or  circles.  Teeth  formed  upon 
these  principles  possess  this  inconvenience :  a  wheel  of  a 
given  pitch  and  number  of  teeth,  say  40,  if  it  be  made  to 
work  correctly  with  a  wheel  of  50  teeth  of  the  same 
pitch,  will  not  work  correctly  with  a  wheel  of  100  teeth  of 
the  same  pitch.  This  is  obvious,  for  the  diameter  of  the 
describing  circle  by  which  the  epicycloid  is  formed  must 
be  made  equal  to  the  radius  of  the  pitch  circle  of  the 
wheel  with  which  the  teeth  are  to  work,  and  will  therefore 
be  twice  as  large  in  the  second  case  as  in  the  first. 

In  the  old  style  of  mill-work,  in  which  the  teeth  of 
wheels  always  consisted  of  wooden  cogs,  this  property 
offered  no  very  serious  impediment,  although,  as  we  shall 
seep  it  introduced  some  complication  of  method ;  but  in  the 
modem  practice  of  making  cast  iron  wheels,  the  objection 
18  a  very  serious  one.  A  founder  must  make  a  new  pat- 
tern of  a  wheel  of  40  teeth  for  every  combination  that  it 


14^  ON   THE  TEETH   OF  WHEELS*        [APPEND. 

may  be  required  to  make  of  this  wheel  with  others,  an^^ 
the  same  for  a  wheel  of  any  other  number.     Besides,  i  ^ 
often  happens  in  machinery,  that  one  wheel  is  required  tc^^ 
drive  two  or  more  whose  number  of  teeth  are  different,  aodL 
in  this  case  the  teeth  cannot  be  correctly  formed  at  all  on 
the  common  principles ;  and  again,  the  perfiscticm  of  ma^ 
dunery  is  impaired  from  the  temptation  to  employ  in  one 
combination  patterns  that  have  been  formed  for  some  other 
combination  very  nearly  the  same ;  for  example,  to  make  a 
wheel  of  40  teeth  that  has  been  formed  to  work  with  one 
of  80,  serve  for  a  required  combination  of  40  with  85. 

It  is  essential,  therefore,  that  the  teeth  of  wheels  should, 
if  possible,  be  so  formed  as  to  allow  a  given  wheel  to  woiIl 
correctly  with  any  other  wheel  of  the  same  pitch.  Now  it 
has  long  been  known  that  involute  teeth  have  this  very  pro- 
perty, but  the  objections  to  these  teeth  on  the  score  of  the 
obliquity  of  their  action  have  operated  fatally  against  their 
introduction*.  I  shall  now,  therefore,  explain  a  method 
of  imparting  to  epicycloidal  teeth  this  properly,  and  thai 
without  making  them  deviate  very  much  frt>m  the  general 
form  that  has  been  established  by  practice. 

To  effect  this,  it  is  merely  necessary  to  employ  a  propo* 
sition  well  known  and  stated  by  almost  every  writer  on  the 
subject,  namely.  If  there  be  two  pitch  circles  touching 
each  other,  then  an  epicycloidal  tooth  formed  by  causing 
a  given  describing  circle  to  roll  on  the  exterior  circumfer- 
ence of  the  one,  will  work  correctly  with  an  interior  epi- 
cycloid, formed  by  causing  the  same  describing  circle  to 
roll  on  the  interior  circumference  of  the  other  t. 

This  proposition  having  been  already  demonstrated,  it 
is  unnecessary  for  me  to  dwell  upon  it  longer  than  to  re- 
mark, that  our  author,  like  all  the  other  writers  on  the 
subject,  has  passed  from  it,  to  recommend  for  practice  that 

*  Vide  Hawkins's  Notes  to  Camus,  page  161. 
t  Vide  Art,  19. 


trEND.  A.^         ON    THE    TEETH    OF    WHEELS.  143 

particular  case  of  it  in  which  the  describing  circle  being 
made  equal  in  diameter  to  the  radius  of  the  pitch  line,  the 
interior  epicycloid  becomes  a  radial  straight  Hoe,  the  in- 
couTOTiences  of  which  practice  I  have  shewn*. 

The  following  corollary  is,  I  believe,  new,  and  consti- 
tutes the  basis  of  the  system  I  propose  to  explain. 

Conilliirt/.  If  for  a  set  of  wheels  of  the  same  pitch,  a 
constant  describing  circle  he  taken,  and  employed  to  trace 
those  portions  of  the  teeth  which  project  beyond  each  pitch 
line  by  rolling  on  the  exterior  circumference,  and  those 
«iuch  lie  within  it  by  roiling  on  its  interior  circumference  : 
then  any  two  wheq^  of  this  set  will  work  correctly  togc- 
ther. 

For,  in  the  first  place,  it  is  well  known  and  can  be 
shewn  from  general  principles,  that  the  portion  of  tooth 
jrithin  the  pitch  line  of  a  driving  wheel,  works  only  with 
the  portion  that  lies  bei/oiid  the  pitch  line  of  its  follower, 
and  that  its  action  is  confined  to  the  approach  of  the  point 
of  contact  to  the  line  of  centres.  After  the  point  of  con- 
t  of  the  teeth  has  passed  that  line,  then  the  case  is  re- 
,  and  the  portion  of  the  driving  tooth  which  lies  be- 
i  the  pitch  line  is  in  contact  only  with  some  part  of  the 
lower*8  tooth  which  lies  within  its  pitch  line. 
Mow  as  a  constant  describing  circle  is  used  fur  the  whole 
it  is  clear  that  the  proposition  will  apply  to  any  pair 
of  wheeU  both  before  and  after  the  teeth  have  passed  the 
line  of  centres,  for  in  each  case  we  have  an  exterior  epicy- 
cloid working  with  an  interior  epicycloid,  and  both  have 
been  drawn  by  the  same  describing  circle,  that  is,  by  the 
ctaistant  circle  of  the  set. 

To  carry  this  scheme  into  practice,  it  only  remains  to 
■ttlc!  the  proper  diameter  to  be  given  to  this  constant  dc- 

'  Vide  Brewsler'fl  Ferguson,  Voi.  II.  p.  223.    CamuG,  p.  27,  or  25  new 


144  ON   THE   TEETH    OF   WHEELS.         [APPEND,    j^ 

scribing  circle,  which  may  be  done  by  considering    tli, 
effect  this  diameter  has  upon  the  form  of  the  tooth. 

Let  Bc  Tif  Fig.  S,  be  a  pitch  circle  whose  centre  is  c, 
then  upon  this  system  the  flank  of  the  tooth,  or  that  por- 
tion which  lies  within  the  pitch  circle,  will  be  an  arc  of  an 

Fig.  2. 


interior  epicycloid  (or  hypocycloid)  mfn  or  mn.  Now  if 
the  describing  circle  be  of  half  the  diameter  of  the  pitch 
line,  the  flank  will  become  a  straight  line  coinciding  with 
the  radius  on.  If  the  describing  circle  be  of  less  than 
half  the  diameter  of  the  pitch  line,  the  flank  mn  will  be 
concave,  and  the  base  of  the  tooth  will  spread ;  but  if  the 
describing  circle  be  of  more  than  half  the  diameter,  the 
flank  mfn  will  be  convex,  and  the  base  of  the  tooth  lessen 
inwards,  a  form  manifestly  unpractical  and  useless.  Hence 
the  describing  circle  must  not  be  greater  than  half  the  dia- 
meter of  the  pitch  line. 

On  the  other  hand,  if  the  diameter  be  too  small,  the 
base  of  the  tooth  will  spread  inconveniently,  and  the  curv- 
ature of  the  exterior  epicycloids  be  injuriously  increased, 
therefore,  on  these  grounds,  it  should  be  made  as  large  as 
it  can  consistently  with  the  limitation  just  stated,  so  that 
we  finally  obtain  this  rule  for  finding  the  diameter  of  the 
constant  describing  circle  for  a  set  of  wheels. 


IPPKVD.  A.]         OS    THE    TEETH    OF    WHEELS.  H5 

Make  it  equal  to  the  radios  of  the  least  pitch  circle  of 
the  stt. 
And  as  pinions  should  never  have  less  than  12  or  14 
!th,  it  would  be  well  to  establish  one  of  these  numbers 
that  least  pitch  circle. 

The  proposition  and  corollary  being  perfectly  general, 
apply  to  racks,  which  must  be  considered  as  very  large 
Is,  and  also  to  annular  or  internal  wheels.  Accord- 
ly,  if  the  constant  describing  circle  be  employed  in 
ring  their  teeth,  they  will  work  correctly  with  any  wheel 
the  set. 

It  rill  be  seen  that  this  system  is  more  easy  of  practice 
the  workman  than  the  old  one.  Every  epicycloid  re- 
es  two  circular  or  rather  segmental  templets,  which  are 
illy  cut  out  of  thin  board.  One  of  these,  which  may 
termed  the  pitch  templet,  has  its  edge  formed  into  an 
of  the  pitch  line  of  the  wheel ;  the  other,  which  re- 
lents the  describing  circle,  and  may  be  called  the  de- 
bing  templet,  has  its  circular  edge  formed  accordingly, 
I  tracing  point  is  fixed  upon  the  circumference  of  the 
ST,  and  the  workman  having  previously  described  an 
of  the  pitch  circle  of  the  wheel  upon  his  drawing 
d,  fixes  the  pitch  templet,  so  that  its  edge  may  coincide 
this  arc,  and  then  causing  the  describing  templet  to 
upoQ  the  pitch  templet,  he  traces  the  arc  of  the  re- 
epicycloid, 

bw  on  the  old  system,  a  set  of  wheels  requires  as  many 
ilets  as  there  are  pitch  circles  in  the  set,  and  also  as 
f  describing  templets,  but  on  the  system  just  explained, 
one  describing  templet  is  needed.  As,  however,  the 
s  of  the  l«eth  within  the  pitch  circles  become  curves 
i)f  straight  lines,  it  is  necessarj'  to  have  concave 
ilets  a<lapted  to  the  pitch  circles,  upon  whose  edges  the 
ibiiig  templet  may  be  made  to  roll  for  the  purpose  of 
the  proper  interior  epicycloid.     The  best  way  is 


D 


14G  ON   THE   TEETH    OF   WHEELS. 


to  make  each  pitch  templet  with  two  edges,  one  cootox  and 
the  other  coDcaTe,  as  in  the  figure,  and  to  write  the  dia- 
meter upon  each  of  them. 


ON   A    POBM  OP   INCKBASBD  STBBITaTH. 

In  a  laige  class  of  machinen-,  the  wheels  constantly 
move  in  the  same  direction,  and  whenever  this  is  the  case, 
it  is  possible  to  increase  the  strength  of  the  teeth  in  a  veiy 
great  d^jee,  by  an  alteration  of  the  common  form  repre- 
sented in  Figure  3. 


Let  AB,  CD,  be  the  acting  iaces  of  the  teeth  of  a  pair  of 
wheels,  of  which  mn,  ks,  are  parts  of  the  pitch  lines- 
Now,  according  to  the  ordinary  practice,  the  backs  of  the 
teeth  would  be  formed  exactly  in  the  same  maimer  as  the 
acting  faces,  as  shewn  by  the  dotted  tines,  and  this  enables 
the  teeth  to  work  backwards  or  forwards  at  pleasure  if  re- 
quired. If,  however,  the  back  is  never  required  to  ac^  the 
strength  of  the  tooth  will  be  nearly  doubled  by  "■■^"g  it 
of  the  form  BegK,  that  is,  by  taking  off  the  portian  tm*, 


'0 


ON    THE    TEETH    OF   WHEELS. 


147 


and  filling  up  the  nook  eng.  Teeth  so  formed  will  clear 
eich  other  quite  as  well  as  those  formed  in  the  usual  man- 
iier,  with  the  advantage  of  a  root  of  nearly  double  extent, 
snd  as  the  acting  faces  remain  of  the  usual  form,  they  will 
fork  together  just  as  the  ordinary  teeth  do.  Strictly 
speaking,  the  back  b  eg  should  be  an  arc  of  an  involute  so 
proportioned  as  to  work  correctly  with  the  corresponding 
b.tck  (if  the  tooth  of  the  other  wheel.  For  then,  as  the 
W'ks  of  the  teeth  would  drive  each  other  truly,  they  are 
Bure  to  clear  each  other  ;  and  besides,  if  the  machinery  be 
made  accidentally  to  run  backwards,  the  teeth  will  still 
work,  although  with  a  considerable  divergent  pressure  upon 
lb  axes'.  It  will  be  quite  near  enough,  however,  to  make 
the  back  an  arc  of  a  circle  described  through  three  points 
Beg',  the  first  of  which,  b,  should  be  taken  a  little  way 
from  the  point  of  the  tooth  in  order  to  blunt  it  slightly  j 
the  second,  e,  on  the  pitch  circle  is  set  off  in  the  usual 
inimner,  so  that  le  may  be  about  i^ths  of  the  pitch  j  and 
the  third  point,  g,  may  be  found  by  dividing  ak  into  five 
I,  and  taking  gK  equal  to  one  of  them.  The  space 
s  required  to  enable  the  point  of  the  corresponding 
b  to  clear  itself. 
3  form  resembles  the  saw-shaped  teeth  which  have 
employed  occasionally  by  mechanists,  for  example, 
»rding  to  Mr.  Reid,  in  his  Horology,  p.  100,)  Lepine, 
,  had  in  some  of  his  watches  the  teeth  and  pinion- 
s  of  a  saw-teeth  form,  hut  I  am  not  aware  that  the 
ntage  of  this  shape  has  ever  been  systematically  shewn, 
J  principle  of  its  formation  given. 

A  divergent  pre^ure  will  do  no  hsnn,  ttecanse  the  kmd  of  mo- 
J  to  which  I  propose  to  adapt  this  form,  never  drives  backwards, 

e  workiog  presiure  i%  upon  it,  but  only  during  some  preyioiia  od- 
mts,  when  the  only  pressure  to  be  overcome  is  that  produced  hy 

T  by  the  friction  of  the  parta  of  the  engine  upon  each  otiier. 
L  2 


148  ON   THE   TEETH    OF   WHEELS.        [APPEND.    A. 

SECTION  11. 

ON    A    PRACTICAL    APPROXIMATION   TO    THE    TRUE   FORM 

BY   ARCS   OF   CIRCLES. 

Although  the  practice  in  the  best  workshops  is  to  de- 
scribe the  shape  of  a  tooth  carefully  with  templets  in  the 
manner  just  described,  yet  this  is  not  done  for  every  tooth 
in  the  wheel  or  pattern ;  on  the  contrary,  having  traced 
the  shape  of  a  single  tooth  in  this  manner,  the  workman 
next  finds  with  his  compasses,  by  trial,  a  centre  and  small 
radius  by  which  an  arc  of  a  circle  can  be  described  that 
will  coincide  as  nearly  as  he  can  manage  to  make  it  with 
the  templet-traced  epicycloid.     Then  having  struck  up(m 
the  face  of  the  rough  cogs  a  circle  concentric  with  the 
pitch  circle,  and  whose  distance  from  it  is  equal  to  that  of 
the  centre  of  his  arc,  he  adjusts  his  compasses  to  the  small 
radius,  and  always  keeping  one  point  in  the  circle  just  de- 
scribed, he  steps  with  the  other  to  each  cog  in  succession, 
they  having  been  previously  divided  into  equal  parts  cor- 
responding  to  the  pitch  and  breadth  of  the  teeth.     Upon 
each  cog  he  describes  two  arcs,  one  to  the  right  and  the 
other  to  the  left,  which  serve  him  as  guides  in  shaping  and 
finishing  the  acting  faces. 

The  portion  of  curve  employed  in  a  tooth  is  so  short, 
that  a  circular  arc  would  be  quite  sufficiently  accurate,  if 
its  centre  and  radius  were  determined  more  correctly  than 
by  this  coarse  mode  of  triaL  This  consideration  induced 
me  to  investigate  the  method  I  am  about  to  describe,  in 
which  the  examination  of  the  nature  and  properties  of  the 
curves  made  use  of  for  teeth  is  entirely  dispensed  with.  I 
have  deduced  a  simple  construction  by  which  a  pair  of 
centres  may  at  once  be  assigned  for  a  given  pair  of  wheels, 
from  whence,  if  arcs  of  circles  be  struck  and  employed  for 


APPENO.  A.']        ON    THE    TEETH    UF    WHEF.tS. 


149 


^M  ttie  working  faces  of  teeth,  they  will  answer  the  purpose  of 

^H  enabling  these  wheels  to  work  correctly  together*. 

^^       I  shall  first  explain  the  methods  that  arise  from  their 

tofijtruction  for  the  use  of  practical  men,  and  then  add  the 

theon-  upon  which  they  are  founded. 
The  working  face  of  each  tooth  may  be  formed  of  one  arc 

of  a  circle,  or  of  two.  In  Fig.  8,  Plate  20,  each  face  is  formed 

»  of  a  single  arc  op,  and  the  resulting  tooth  has  considerable 
snali^  to  the  involute  tooth,  and  hke  it  has  the  fault  of 
acting  with  rather  too  much  obliquity  j  but  the  mode  of 
describing  it,  on  the  other  hand,  is  exceedingly  simple,  and 
any  two  wheels  whose  teeth  are  thus  formed,  will  work  cor- 
rectly together.  For  small  teeth,  I  am  inclined  to  think 
tliis  method  would  answer  the  purpose  very  well.  It  only 
remains  to  show  how  the  centre  p  and  radius  i't  of  the  arc 
oTp  is  to  be  determined. 

Let  A  T  be  the  radius  of  the  pitch  circle  of  the  proposed 

Hvlieel.     Upon  at  describe  a  semicircle  tpa,  and  from  t 

Hlbt  off  t  p  equal  to  one  quarter  of  the  radius ;  then  will  i> 

^De  the  centre  from  which,  if  an  arc  op  be  described  through 

T,  the  required  side  of  the  tooth  will  be  obtained. 

Or,  construct  a  bevil  in  brass,  of  which  the  angle  at  t 
&hall  be  exactly  equal  to  75°  Sty,  and  graduate  the  side  tp 

*  Eut(ir,  in  his  secooJ  paper  on  the  teeth  of  wheels,  (N.  C.  Pet.  XI. 

_'ii9,)  li&s  with  his  usual  ftbility  investigated  the  proper  ciutck,  by  examin- 

■  I'j  the  rchttion  between  their  radii  of  curvature  at  every  point.     This  me- 

^iiod  hoB  uMurally  fondui^ted  him  to  results   of  a  similar  nature  to  those 

li  I  have  given  in  tlie  following  pages,  and  he  Ruggests  that  a  smaJl  arc 

(die  circle  of  curvature  would  suffice  in  practice  for  the  forms  of  teetb. 

II  given  some  geometrical  constmctionB  for  this  purpose,  and  has  then 

jn  finally  to  recommend  the  involute  oa  the  best  cuire,  this  paper 

a  fact,  the  first  in  which  that  curve  ts  pointed  out  as  possessing  the 

d  pmperties.     To  Euler,  then,  belongs  the  merit  of  first  suggesting 

p  sabiititudon  of  an  arc  of  the  circle  of  cun-atiire  for  the  real  curve,  a 

I  wkicb  bos  been,  as  for  as    I  know,  neglected  by  every  succeeding 

"writer.     This  may  perhaps  be  utlribiitcd  to  the  abstruse  manner  in  whicb 

b<>  hu  treated  the  subject. 


150  ON   THE   TEETH   OF   WHEELS.        [APPEND.  A. 

into  a  scale  of  quarter  inches,  as  in  the  figure.  Apply  the 
plain  side  of  this  bevil  to  the  radius  at  of  the  proposed 
wheel,  and  its  point  t  to  the  pitch  circle ;  read  off  the  length 
of  the  radius  a  t  in  inches  upon  the  reduced  scale  t  p,  and 
the  point  p  so  indicated  will  he  the  centre  of  the  tooth  as 
before.  Thus  in  the  figure,  at  is  four  inches,  and  p  is 
found  at  4  upon  the  scale. 

When  the  side  of  the  tooth  is  formed  of  two  arcs  of  cirdeB, 
the  forms  shown  in  Figs.  9  and  10  are  obtained :  these  re» 
present  the  same  teeth  in  different  relative  positions.  In 
Fig.  9  the  tooth  abc  is  approaching  the  line  of  centres  ab, 
and  in  Fig.  10  the  same  tooth  abc  is  retiring  from  it.  The 
portion  of  tooth  ab  which  lies  within  the  pitch  circle,  is 
described  from  a  centre  p,  Fig.  9 ;  and  the  portion  be  which 
lies  beyond  the  pitch  circle  is  described  from  a  centre  p,  fig. 
10.  The  resulting  form  is  a  very  strong  one,  possessing 
the  property  that  any  two  wheels  of  a  set  will  work  to- 
gether. Any  practical  man  may  convince  himself  of  the 
degree  of  accuracy  with  which  this  is  effected,  by  describing 
according  to  this  method,  on  a  large  scale,  (say  six  inches 
pitch,)  a  pinion  of  twelve  or  fourteen  teeth,  and  a  few  teeth 
both  of  a  wheel  of  fifty  and  of  a  rack.  These  teeth  may  be 
cut  out  of  thin  board,  and  it  will  be  found  that  any  two  ci 
them  will  work  correctly  together  with  a  degree  of  pre- 
cision amply  sufficient  for  practice.  To  facilitate  the  descrip- 
tion of  teeth  as  much  as  possible,  I  have  thrown  the  system 
into  the  form  of  an  instrument,  which  I  have  termed  an 
Odontagraph,  and  which  any  one  may  make  for  themselves 
out  of  a  sheet  of  card  paper,  by  observing  the  following  in- 
structions*. 

FED^  Fig.  11.  represents  this  instrument  on  a  scale  oi 
one  quarter  of  the  originaL  The  angle  d  <f  is  exactly  J5% 
and  the  side  k^f  is  graduated  into  a  scale  of  half  inches, 

*  Those  who  arc  not  disposed  to  take  the  trouble,  may  obtain  it 
plcte  of  Mos^n^.  Holtzupfcl  of  Charing  Cross. 


APPEND,  A.]         ON    THE    TEETH    OF    WHEELS.                           151 

each  half  inch  being  divided  into  ten  parts.     The  half 

inches  are  numbered  from  zero  at  /,  both  ways  towards  the 

extremity  of  the  scale,  0,    10,  20, up  to  about  200 

upwards,  and  40  downwards. 

Upon  the  plain  surface  of  the  card  are  placed  the  tables 

which  follow: 

TABLES  SHEWING  THE  PLACE  OF  THE  CENTBES 

UPON  THE  SCALE. 

1         CENTRES  FOR  TEETH  WITHIN  THE  PITCH  CIRCLE. 

'                                                    PnCB   IN   MCHEa   AND  tlL^IS. 

i 

I 

i 

5 

1 

H 

n 

1} 

2 

n 

2i 

3 

H 

13    I32 

48 

6* 

80 

96 

129 

160 

193 

225 

257 

289 

321 

386 

450 

U    |l7 

26 

35 

13 

52 

69 

87 

104 

121 

139 

156 

173 

208 

242 

\5     .12 

18 

25 

31 

37 

49 

62 

7* 

86 

9S 

123 

148 

173 

16 

10 

15 

20 

25 

30 

40 

50 

59 

69 

79 

89 

i)9 

1^1 

IT 

8 

13 

17 

21 

25 

34. 

42 

SO 

59 

67 

75 

84 

101 

117 

IS 

7 

n 

15 

19 

22 

30 

37 

45 

52 

59 

67 

74 

89 

104 

19 

10 

13 

17 

20 

27 

35 

40 

47 

54 

6(1 

67 

80 

94 

20 

e 

9 

12 

16 

18 

25 

31 

37 

43 

49 

56 

62 

74 

80 

22 

5 

8 

11 

U 

16 

22 

33 

39 

43 

49 

54 

65 

76 

a* 

7 

10 

12 

15 

20 

25 

30 

35 

40 

45 

49 

59 

69 

26 

... 

9 

11 

14 

18 

23 

27 

32 

37 

41 

46 

55 

64 

28 

» 

6 

13 

22 

26 

30 

35 

40 

43 

52 

60 

30 

... 

8 

in 

12 

17 

21 

25 

29 

33 

37 

41 

49 

58 

33 

... 

9 

11 

16 

19 

23 

26 

30 

34 

38 

45 

.13 

to 

... 

5 

7 

15 

18 

21 

25 

28 

32 

35 

42 

49 

eti 

= 

e 

8 

9 

13 

15 

19 

22 

25 

28 

31 

37 

43 

80     ... 

4 

7 

12 

17 

20 

23 

26 

29 

35 

41 

100 

"s 

11 

14 
13 

16 

19 

22 
21 

25 
24 

28 
27 

34 
32 

39 
38 

m 

5 

wt. .  2 

e 

7 

10 

12 

15 

17 

20 

22 

25 

30 

34 

1         CENTRES  FOR  TEETH  OUTSIDE  THE  PITCH  CIRCLE. 

'                                         f,t™  in  iscbm  akd  p*mT«. 

Number 
■■fTwrt. 

i 

8 

1 

i 

i 

1 

U 

H 

'? 

2 

2J 

2! 

3 

3i 

18 

7 

2 

2 

3 

4 

5 

e 

~ 

9 

10 

11 

12 

15 

17 

15 

3 

7 

A 

10 

11 

12 

14 

17 

19 

V 

4 

'5 

B 

8 

9 

11 

12 

14 

15 

18 

21 

^tm/t 

3 

'4 

7 

9 

10 

12 

14 

16 

18 

21 

25 

1 

6 

8 
9 

10 
n 

11 
12 
13 

14 

13 

14 
15 

16 

15 
16 
17 
18 
19 

17 
18 
19 

20 
21 

19 
20 
21 
22 
23 

23 

25 
26 

27 

26 
29 
30 
31 
33 

's 

7 

B 

« 

t; 

* 

i'ft 

12 

!5 

17 

20 

22 

25 

90 

34 

i 

L 

L9S 


OS   THK 


One  example  viHe^iiiai 
meiit*     Let  it  be  reqoiRd  id 
fer  a  wheel  of  29  ceech  of -5 

Describe  an  arc  t/  of  tfae  jwj^iici 
offnpoD  it  T/,  equal  to  tfaepDciL 
radial  lines  bt,  b/.     For  the 
applj  the  slant  edge  of  the 
placing  its  extrenntr  /  oo  the  pitek 


of  a  tao& 


Gorde,  and  ■! 
me;  drnr 
pifim  GRv 
^e  rsSal  Bat  nt, 
IB.  the  apnSi 


In  the  table,  headed  Centre*  J^  teedk  ntibn  Ae  fUk 
circle^  look  down  the  oofamm  of  -5  isA  pbcft^  and  oppoHte 
to  30  teeth,  wfaicb  is  the  nearest  namber  n>  dac  rehired, 
win  be  foond  the  mnnber  40-  The  pant  r,  indkatsdci 
the  drawing  board  br  the  pmitWw  of  this  number  on  die 
scale  of  equal  parts  marked,  5ca/e  o^cmlrer  ftfteeA  rittts 
^'/cA  circle^  is  the  centre  reqaired,  from  which  the  arc  e/ 
most  be  drawn  with  a  radios  re. 

The  centre  for  the  arc  de^  which  lies  oatdde  the  pitdi 
circle,  is  formed  in  a  manner  predselT  similar,  br  iqiplyiDg 
the  slant  edge  of  the  scale  to  the  radial  line  bt.  The 
number  ^21  obtained  from  the  table  of  Centres  fior  teeth 
etrtside  the  pitch  circle  will  indicate  the  positicHi  of  this 
centre  upon  the  ^Scale  of  centres  Jbr  teeth  outside  the  pitch 
circle^  namely  at  r. 

The  radius  of  the  wheel  may  be  found,  by  help  of  the 
following  table  and  rule.  Multiply  the  number  correspond- 
ing to  the  given  pitch  in  this  table  by  the  number  of  teeth 
required,  the  product  will  be  the  radius  of  the  pitch  circle  in 
inches  and  decimals.  Thus,forawheelof^  teeth  of  3  inches 
pitch,  multiply  '4774'  by  29»  and  the  radius  is  13*84  inches. 


Pitch. 

Facton. 

!     PHch. 

Facton. 

3J 

•5570     ' 

■ 

•1989 

3 

•4774 

1 

•1591 

'          91 

•3979 

1 

•1193 

2: 

•3581 

!         1 

•0994 

2 

•3183 

1 

•0795 

If 

•2785 

1 

•0597 

l| 

•2387 

1 

•0398 

VPEND.  A.]        ON   THE    TEETH    OF    WHESI 

The  curve  def,  Fig.  11,  is  also  true  for  an  annular  wfieel 
t  the  same  number  of  teeth,  ^becoming',  of  course,  the 
point  of  the  tooth,  and  d  its  root.  For  a  Rack,  the  pitch 
T  t  will  be  a  straight  line,  and  b  /,  b  t  bo  drawn  perpen- 
dicular to  it,  at  a  distance  ironi  each  other  equal  to  the 
jitch.  The  numbers  for  pitches  not  inserted  in  the  table, 
wy  bo  obtained  from  the  column  of  some  other  pitch,  by 
liirect  proportion.  Thus  for  4  inch  pitch,  by  doubling  the 
imb^-s  in  the  column  of  the  2  inch  piteh,  for  1^  by 
fmhling  ^\,  and  so  on ;  or  if  the  difference  be  small,  the 
nlamn  belonging  to  the  nearest  piteh  may  bo  employed, 
nthout  a  serious  error  ;  or  more  accurately  a  number  may 
il  taken  half  way  between  those  given  in  the  two  nearest 
jBtumns. 

I  No  tabular  numbers  are  given  for  twelve  teeth,  for  with- 
B  the  piteh  circle  such  teeth  are  bounded  by  radial  lines. 
I  But  without  using  the  Odontagraph,  the  geometrical 
istruction  shown  in  Figs.  9  and  10  may  be  employed, 
s  must  be  of  course  drawn  to  the  real  size  of  the  wheels 
D  question. 

Let  A  B  be  the  centres  of  a  pair  of  wheels,  r  the  point  of 
oontingcncc  of  their  piteh  circles;  through  t  draw  ktk, 
making  on  angle  of  15"  with  the  line  of  centres,  and  hi- 
sected  in  T;  also  draw  pt  perpendicular  to  ktk,  tk  may 
be  of  any  length  less  than  the  least  radius  of  the  piteh 
circles-  There  are  thus  obtained  two  points  k,  one  near  to 
the  right  hand  centre  b,  and  the  other  to  the  left  hand 
centre  s.  The  first  is  thus  employed  in  Fig.  10,  to  obtain 
the  arcs  be,  ef.  Join  hk  and  produce  it  to  q;  join  ak  in- 
tersecting QT  in  p.  Set  off  T  »  equal  to  half  the  pitch,  and 
with  centre  f  and  radius  m  describe  the  arc  be  outside 
the  pitch  circle  of  the  left  hand  wheel,  and  with  centre  <i 
and  radius  qm  describe  the  are  e/"  within  the  pitch  circle  of 
!  opposite  wheel,  whose  centre  of  motion  is  b  ;  then  will 
!  arcs  work  truly  together.  In  like  manner  Fig.  9. 
L  and  produce  it  to  meet  the  line  tqp  in  p ;  join  i 


154  ON    THE    TEETH    OF   WHEELS.        [APPEND.  i%. 

intersecting  the  same  line  in  q.  Set  off  Tm  equal  to  haJf 
the  pitch,  and  with  centre  p  and  radius  Ftn  describe  thAt 
portion  of  the  tooth  ab  which  lies  within  the  pitch  cirdle 
&T)  and  with  centre  q  and  radius  Qm  describe  the  tootli 
ed^  which  lies  beyond  the  pitch  circle  eT. 

These  rules  must  be  observed  in  both  pases,  namely,  thAt 
the  half  pitch  Tn,  Tm  is  always  set  off  on  the  opposite 
side  of  the  line  of  centres  to  the  centres  pq  of  the  teeth  ; 
and  that  the  centres  of  the  concave  flanks  within  the  pitcl 
circle  are  obtained  by  joining  the  centre  of  the  pitch  circle 
with  that  Ky  which  lies  nearest  to  it,  and  producing  the 
line  to  meet  tqp  :  but  that  the  centres  of  the  convex  teeti 
beyond  the  pitch  circle  are  obtamed  by  joining  the  centre 
of  the  pitch  circle  with  that  k,  which  is  most  remote  from  it 

Any  two  wheels  in  which  the  length  of  kt  is  the  same, 
will  work  truly  together. 


ON   TBBTH   WORKING   WITH   TRUNDLES  OB  RADIAL   PLANKS. 

The  particular  applications  of  the  general  construction 
which  I  have  given,  apply  only  to  complete  sets  of  wheels 
working  together,  and  it  may  be  as  well  to  shew  its  use  in 
obtaining   teeth   adapted   to  work  with   trundles   or  pin 
wheels,  as  well  as  teeth  in  which  the  flank  is  a  radial  line 
as  in  the  common  form.     The  diagram  of  Fig.  12,  Plate 
20,  must  be  drawn  of  the  full  size  for  any  required  wheel, 
A  and  B  are  the  centres  as  usual,  fg  and  hk  arcs  of  the 
pitch  circles.     Upon  the  radius  of  the  trundle  at  describe 
a  semicircle,  upon  which  set  oflF  from  t,  tp  equal  to  the 
pitch.     Draw  p  t  q  and  let  fall  a  perpendicular  b  q  upon  it 
from  B,  intersecting  it  in  q.     If  the  point  p  be  taken  for 
the  centre  of  the  stave  or  pin,  an  arc  mn  described  from 
Q  and  touching  the  stave  in  m,  will  be  the  side  of  the  tooth 
required. 

If  the  flanks  of  the  teeth  are  to  be  radial  lines,  then  the 
portions  lying  without  the  pitch  circle  may  be  arcs  of  cir- 


.IPPEND.  A.]        OS    THE   TEETH    OF    WHEELS.  155 

ties  found  thus.  (Fig.  14.)  a  and  b  are  the  centres  ;  fg, 
ilk,  arcs  of  the  pitch  circles  as  before.  Upon  at  describe 
an  entire  circle,  and  upon  its  circumference  from  a  and  t 
set  off  A  J,  T7n  equal  to  each  other  and  to  about  three 
quarters  of  the  pitch  ;  join  bz  and  through  mi  draw  nnq. 
intersoctiog  bz  in  a;  then  an  arc  described  from  centre  Q 
ad  struck  through  m,  will  be  the  curved  face  of  the  tooth 
for  BT,  and  this  will  work  with  the  radial  flank  of  the 
loolh  of  A  T.  To  find  the  curved  face  of  the  latter  tooth 
Biakc  a  similar  diagram,  in  which  at  and  bt  exchange 
places. 


ON   CUTTERS. 


The  Odontagraph  is  also  applicable  to  the  obtaining  a 
larrect  form  for  the  cutters  used  in  shaping  the  teeth  of 
tintal  wheels.  The  form  of  the  cutter  is  that  of  the  space 
between  two  teeth,  and  in  order  to  shew  the  nature  of  the 
fhange  of  form  required  for  different  t^eth  as  well  as  the 
general  form  itself,  I  have  in  Figure  13,  Plate  20,  drawn 
«ith  accuracy,  and  on  a  large  scale,  the  teeth  proper  to  the 
fiFo  extreme  cases  of  a  pinion  of  12  on  the  one  hand,  and 
3  rack  on  the  other,  and  have  applied  these  two  together, 
50  that  the  central  line  of  the  spaces  shall  coincide,  and 
I  ibiu  bring  the  shapes  of  the  cutters  into  direct  compari- 

Bow  between  these  two  lie  all  the  forms  that  are  re- 

1  for  any  number  of  teeth  from  12  to  a  rack,  or  the 

»t  possible  wheel ;  but  in  making  a  set  of  cutters,  for 

pitches  especially,  it  is  by  no  means  necessary  to 

ke  one  for  every  number,  as  the  forms  for  numbers  that 

■  close  together  are  so  nearly  ahke  that  the  errors  of 

irkmanship  would  entirely  destroy  the  difference. 

■  The  \-ariation  of  form  however  is  much  less  among  high 

Bobers  than  in  low  ones.     For  example,  the  difference  of 

between  a  cutter  for  150  teeth,  and  one  for  300,  is 

k  greater  than  that  between  cutters  for  16  and  17  teeth. 


156 


ON   THE   TEETH    OF   WHEELS.         ^APPEND,  A. 


This  being  the  case,  it  appeared  worth  while  to  investi- 
gate some  rule  by  which  the  necessary  cutters  could  be  de- 
termined for  a  set  of  wheels,  so  as  to  incur  the  least  pos- 
sible chance  of  error.  To  this  eflFect  I  have  calculated,  by 
a  method  sufficiently  accurate  for  the  purpose,  the  follow- 
ing series  of  what  may  be  termed  equidistant  values  (tf 
cutters ;  that  is,  a  table  of  cutters  so  arranged,  that  the 
same  difference  of  form  exists  between  any  two  consecutive 
numbers. 


TABLE   OF   EQUIDISTANT   VALUES   FOR  CUTTEB8. 


No.  of 
Teeth. 


Raek. 


300 


IfiO 


100 


76 


as 


uu 


ISx 


11 


This  will  be  a  guide  in  the  selection  of  the  wheel  to 
which  each  cutter  shall  be  accurately  adapted  after  it  has 
been  determined  how  many  are  necessary  in  a  set  For 
example,  if  a  single  cutter  were  thought  sufficient  for  a  set  of 
very  small  wheels,  it  had  better  be  accurately  adapted  to  teeth 
of  25,  for  that  value  is  intermediate  between  the  two  ex- 
tremes. If  three  cutters  are  to  suffice  for  the  whole  set* 
then  76,  25,  and  15  must  be  selected,  of  which  the  cutter 
76  may  be  used  for  all  teeth  from  a  rack  to  38,  the  cutter 
25  from  38  to  19»  and  the  cutter  15  from  19  to  IS,  and  so 


on. 


It  appears  from  the  figure  that  the  greatest  diffisrence  (tf 
form  is  at  the  apex  of  the  tooth,  (that  is,  at  the  base  of  the 
cutter,)  and  amounts  to  *25  inch  in  2  inch  pitch ;  from  this 
the  difference  may  be  ascertained  for  any  smaller  pitdi, 
and  as  many  cutters  interposed  as  the  workman's  notion  of 
his  own  powers  of  accuracy  may  induce  him  to  think  ne- 
cessary. 

Thus  if  the  hundredth  of  an  inch  be  his  limit  of  aoca- 
racy  in  forming  cutters,  and  he  is  making  a  art  far  Jvlf 


APPESD.  A.]        ON    THE    TEETH    OF    WHEELS.                            15?            ^^^H 

inch  pitch,  where  the  difference  of  form  is  ^  x  -25  or  -06             ^^^| 
nearly,  then  half  a  dozen  cutters  will  be  sufficient,  and             ^^H 
these  must  bo  made  as  nearly  as  possible  to  suit  the  wheels             ^^H 
nf  IJO,  ^0,  30,  21,  16,  13.                                                                         ^H 
1        The  following  table  contains  a  selection  of  numbers  for            ^^^| 
Afferent  case:;,  which  may  save  trouble.                                            ^^^| 

TABLB  OP  CUTTSBS.                                                                           ^^^^H 

Witrf 

» 

W 

1 

1 

7» 

M 

It 

1 

1 

lOU 

M 

*> 

14 

_ 

1 

1 

.« 

H> 

so 

II 

dm 

1 

1 

mo 

«7 

to 

»■ 

IS 

1 

ICI 

M 

■n 

M 

33 

^S|S| 

1 

u 

lm 

m 

43 

a  14 

13 

1 

u 

30O 

iw 

100 

70 

«to 

36 

SO 

3D' IB 

4a 

Mia 

SiljS 

»• 

^^ 

** 

1=0 

Hw 

IS, 

uai 

4  30 

^'r 

.;,,»| 

^lien  the  numbers  have  been  selected,  the  Odontagraph             ^^H 
nay  be  employed  to  draw  the  figure  of  the  cutter  corre-             ^^H 
'ponding  to  each  wheel,  either  oa  the  same  scale  as  the  pro-              ^^H 
posed  cutter,   or  on  a  much  larger  scale,  which  may  be             ^^^^ 
^Aerwards  reduced  proportioQally.                                                      ^^^| 

SECTION  in.                                                   ^^H 

THEOBY    OF   THE    PRECEDING   CONSTRUCTIONS.                               ^^H 

We  must  first  examine  the  nature  of  the  motion  which             ^^| 
1    i*  proclucefi  by  the  pressure  of  one  circular  ai-e  upon  another             ^^H 
■^n  disposed  so  as  to  work  in  the  manner  of  teeth.                        ^^| 
^■lut  AB,  Fig.  4.,  he  two  centres  of  motion,  A-mtj  a  piece             ^^H 
^^>nwl  into  a  circular  arc  described  from  a  centre  p,  and               ^^B 
H^*ble  of  revolvuig  round  a  ;  oyip  in  like  manner  a  cir-                      1 
^■bIut  arc  described  from  q,  and  capable  of  revolving  round                      1 
^K  now  if  the  arc  Um/i  be  made  to  press  against  OMp,  so                      1 
^H^eoraiuuaicatc  rotation  to  it  round  b,  the  line  pq,  join-                     M 

158 


ON   THE   TEETH   OF  WHEELS.        [aPPEND^ 


Fio.  4. 


ing  the  centres  of  the  arc  will  necessarily  always  pas^ 
through  the  point  of  contact  m»  and  will  be  of  a  constant 
length  equal  to  the  sum  of  the  radii,  so  that  in  fact  the 
motion  will  be  exactly  the  same,  if  for  the  circular  arcs  a 
link  p  Q  be  substituted,  which  length  is  equal  to  the  sum  of 
the  radii  pm,  qm,  and  which  is  jointed  to  the  revolving 
pieces  at  p  and  q,  the  places  of  the  centres. 

This  also  shews  that  a  change  of  the  actual  lengths  of 
the  radii  pm,  qm,  will  not  aflPect  the  motion,  so  long  as  the 
distance  of  the  centres  is  constant,  for  that  whether  the 
circular  arcs  had  been  struck  through  m  or  m',  or  even 
through  a  point  m^'  beyond  the  centre  q,  the  system  would 
still  have  been  equivalent  to  the  link  pq,  jointed  to  the 
arms  ap,  bq. 

It  is  only  necessary  then  to  examine  the  motion  of  this 
simple  system  of  rods,  and  then  to  explain  how  it  may  be 
employed  in  forming  the  teeth  of  wheels. 

Let  the  rod  a  p.  Fig.  5,  be  moved  into  a  new  position 
A/),  its  extremity  will  carry  with  it  the  end  of  the  link  pq, 
and  communicate  through  it  a  motion  to  the  arm  b  q,  by 
which  it  will  be  driven  into  the  new  position  b  q ;  and  it  is 
necessary  to  know  the  relative  value  of  this  motion  to  that 
of  AP,  which  produced  it 

Now  this  relation  is  continually  changing,  but  its  value 


APPEND,  A.3        ON  JTHB   TEETH   OF   WHEELS. 


159 


Fig.  5. 


at  any  instant  may  be  thus  determined.  The  rod  p  q  during 
its  motion  may  be  considered  as  always  turning  round  some 
centre  or  other  in  space,  although  the  relative  position  of 
that  centre  to  it  is  continually  shifting.  Produce  the  arms 
AP,  BQ  in  the  requisite  directions  to  meet  in  k,  then  will 
this  point  k  be  the  momentary  centre.  For  as  the  ex- 
tremity p  moves  round  the  centre  a,  the  direction  of  its 
motion  at  starting  from  p  must  be  perpendicular  to  ap, 
therefore  the  momentary  centre  will  lie  somewhere  in  a  p 
produced.  In  like  manner  the  initial  motion  of  the  other 
extremity  q  must  be  perpendicular  to  b  q,  and  the  moment- 
ary centre  must  also  lie  somewhere  in  the  direction  of  b  q  : 
therefore  it  must  be  in  the  intersection  k  of  the  two  lines 
AP  and  bq  produced.  But  since  the  rod  pq  turns  on  the 
momentary  centre  k,  the  direct  motion  of  p  and  q  are  to 
each  other  at  any  given  instant  as  their  radial  distances 
firom  K,  that  is,  as  pk  to  qk,  which  is  true,  whether  we 
eonaider  them  as  the  extremities  of  the  rod  p  q  or  of  the 
ndii  AF,  bq;  also  the  angular  motions  of  the  latter  will 
lie  firand  by  dividing  these  direct  motions  by  their  re- 
qieclive  radii  i  therefore  we  have, 


160  ON   THE   TEETH   OF  WHEELS.       [aPPENDV   A, 

Angular  moti<m  of  p  round  a:  angular  motion  o£'  q 

J  PK      QK 

round  b  : :  —  :  — . 

AP       BQ 

Draw  KL,  AM,  BK,  perpendicular  to  pq.      Then  we 

have 

PK  :  AP  ::  kl  :  am  by  similar  triangles  kpl  ;  apm 

bq:qk::bn:kl bqn;klq 

at:bt::am:bn atm;tbn 

and  compounding  these  three  proportions  we  obtain 

PK      QK 

—  :  -1-  ::  BT  :  AT 

AP      BQ 

that  is  to  say,  the  angular  motion  of  the  arms  are  to  eacb 
other  at  any  moment  inversely  as  the  segments  into  which 
the  direction  of  the  link  divides  the  line  joining  the  centres 
of  motion,  or  line  of  centreSj  as  it  is  usually  termed.   K 
now  it  happens  that  when  the  link  p  q  moves  into  its  new 
position  py,  very  near  to  the  first,  this  second  position  in- 
tersects the  first  in  a  point  l  above  (or  below)  the  line  of 
centres,  as  in  the  figure ;  then  the  ratio  of  the  segments 
A  T,  B  T  will  be  altered  into  that  of  a  <,  b  ty  consequently  the 
ratio  of  the  angular  motion  will  be  an  increasing  or  de- 
creasing ratio,  as  the  case  may  be.     But  if  the  point  l 
coincide  with  the  line  of  centres,  this  ratio  will  for  the  mo- 
ment remain  constant. 

Now  a  little  consideration  will  show  that  the  point  of  in- 
tersection between  two  successive  positions  pq,  pq  of  the 
link  must  be  at  the  place  where  the  perpendicular  from  k 
falls  upon  it.     For  as  k  is  the  momentary  centre  of  motion 
of  this  link,   the  extremity  l  of  the  perpendicular  will 
begin  to  move  in  a  line  at  right  angles  with  it,  and  conse- 
quently will  remain  in  the  direction  of  the  first  position  pq 
when  the  link  has  passed  into  the  second  p  9,  that  is  to  say, 
it  will  be  the  point  of  intersection  of  the  two  positions ; 
when,  therefore,  the  rods  are  in  such  a  position   that  the 
perpendicular  from  k  meets  the  link  p  q  in  the  line  of  cen- 
tres, the  ratio  of  the  angular  motions  of  a  p  and  b  q  is  con- 


PI'EMD.  A.]  OH   THE    TEETH    OF    WHEALS. 

Slant:  and  if  in  this  state  of  the  systpm  the  pniuts  p  and  u 
bo  employed  (as  in  Fig.  4)  as  centres  from  whence  short 
arcs  are  drawn  througli  any  common  point  w,  and  applied 
ii&  leeth,  these  arcs  will  manifestly  drive  each  other  cor- 
rfctly  when  in  the  exa«t  relative  position  described,  and 
Teiy  nearly  so  when  removed  to  a  short  distance  on  each 
side  of  it,  which  is  the  thing  required*. 

Now  these  relative  positions  of  p  and  q  may  be  deter- 
mined by  a  simple  construction  founded  upon  the  necessary 
i*incidence  of  l  with  x. 

Let  A  and  b.  Fig.  G,  be  the  given  centres  of  motion,  a  u 
'lie line  of  centres  divided  in  t,  so  that  the  segments  at,  ar 
shall  have  the  ratio  of  the  required  motions ;  or  in  other 
fnrds,  let  T  be  the  point  of  contact  of  the  pitch  lines. 
[)raw  ptq,  making  any  angle  with  ab,  and  through  t 
ilraw  TK  perpendicular  to  it.  Upon  ptq  assume  a  point  p 
as  a  centre,  from  whence  the  circular  arc  or  tooth  belong- 
ing to  A  is  to  be  drawn.  Then,  to  find  the  corresponding 
ccDlre  for  ii,  join  ap,  and  produce  it  to  meet  tk  in  k,  join 
Kb  and  produce  it  to  meet  ptq  in  q.  Then  will  q  be  the 
point  required,  which  will  appear  by  comparing  this  dia- 
smm  with  Fig.  5. 

If  the  point  p  had  been  taken  at  p',  so  that  the  angle 
at't  were  less  than  a  right  angle,  then  the  line  p'a  would 
have  intersected  t  k  in  a  point  k'  on  the  other  side  of  t, 
and  this  would  have  thrown  q  to  n'  nearer  to  t. 

Again,  V  might  have  been  assumed  on  the  other  side  of 
i  B  as  at  p",  but  then  the  driving  arc  struck  through  m 
Would  have  been  concave.  It  is  not  worth  while  to  ex- 
amine all  the  cases  that  arise  from  the  different  relative 
positions  of  the  points ;  I  shall  merely  show  those  that  are 
able  to  practice. 

to  rcniork,  tlittt  u  more  direct  an<I  simple  de- 
migbt  have  been  given  Ly  employing  in- 
desirous  of  avoiding  ill  a  practical  paper. 


162 


ON  THE   TEETH   OF  WHEELS.        [APPEND.  A. 


Fig.  6. 


The  side  of  the  tooth  may  be  formed  either  of  a  single 
arc  or  of  two.     As  the  are  is  only  an  approximation,  and 
is,  strictly  speaking,  only  exact  at  one  point  of  the  action, 
it  will  be  better  to  adopt  a  figure  composed  of  two  arcs  of 
circles,  as  we  obtain  two  exact  points ;  but  in  that  case  one 
arc  should  be  concave  and  the  other  convex,  in  order  to 
facilitate  their  jimction  and  produce  a  wider  base ;  and 
thus  a  figure  is  formed,  as  we  have  seen,  very  near  to  that 
usually  adopted,  the  convex  arc  being  of  course  given  to 
that  part  which  lies  outside  the  pitch  circle,  and  the  con- 
cave to  that  which  extends  within  it. 

The  angle  atp  is  arbitrary,  and  its  value  may  therefore 
be  determined  from  other  conditions  than  those  already 
stated.  If,  however,  it  be  made  a  right  angle,  it  is  clear 
that  the  points  p  and  q  vanish  by  coinciding  with  x  j  and 
if  it  be  made  a  little  less  than  a  right  angle,  the  points  p 


'•] 


THIC    TRETH    OF    WHEELS. 


Ui3 


d  H  are  thrown  so  near  to  t  that  the  radii  by  which  the 
Kuxs  are  struck  become  too  short,  and  the  points  of  the 
teeth  too  much  rounded  off. 

On  the  other  hand,  if  the  angle  a  t  P  is  made  too  acute,  the 
action  of  the  teeth  upon  each  other  at  the  moment  of  passing 
the  lino  of  centres  and  elsewhere  becomes  very  oblique, 
and  an  injurious  pressure  is  thereby  thrown  upon  their 
axes.  By  various  trials  I  have  fixed  upon  75°  as  the  value 
of  the  angle  which  appears  to  avoid  these  two  extremes, 
and  have  accordingly  employed  it  in  the  construction  of 
the  Odontagraph. 

Again,  the  position  of  the  point  vi,  through  which  the 
sns  are  to  be  struck,  is  also  arbitrary,  and  must  be  deter- 
mined by  considering  which  point  of  the  action  we  wish  to 
make  the  correct  point.  If  the  teeth  consist  of  a  single 
flrc  each,  the  correct  point  may  be  fixed  at  the  moment  of 
passing  the  line  of  centres,  and  therefore  the  arcs  must  be 
llnick  through  the  point  t  ;  but  if  the  side  of  the  tooth 
lie  formed  of  two  arcs  joined,  one  lying  within,  the  other 
ItftHid  the  pitch  line,  then  the  action  of  one  of  them  will 
b  confined  to  the  approach  of  the  point  of  contact  of  the 
pth  to  the  line  of  centres,  and  the  action  of  the  other  to 
I  recess  from  that  line,  and  m  must  be  assumed  upon 
a  principle  that  the  correct  pouit  of  each  arc  shall 
^  nearly  in  the  middle  of  its  action,  the  mode  of  doing 
ich  will  appear  presently. 


10   DEBCBIBE   T 


I   aiNQLH   ABC. 


If  the  side  of  the  tooth  consist  of  a  single  arc,  the  sys- 
tem may  be  made  exceedingly  simple,  for  as  the  distance 
■'!"  the  point  k  from  t  is  arbitrary,  when  the  points  p  or  a 
■  •.n:  not  given,  suppose  it  to  be  taken  at  an  infinite  distance, 
1  (Fig.  ())  ATK  and  qbk  will  become  parallel  to  tk, 


164 


ON   THE   TEEtH   OF   WHEELS.        [aPP£ND»  A. 


and  perpendicular  to  ptq,  which  shews  that  if  lines  be 
drawn  from  a  and  b  (Fig.  7)  perpendicular  to  ptq,  the 
points  p  and  q  will  be  centres,  whence  if  arcs  be  drawn 
through  some  common  point  m,  or  rather  in  this  case  t, 
these  arcs  will  drive  each  other  correctly. 

Fig.  7. 


If  the  angle  p  t  a  remain  constant  for  a  set  of  wheels  of 
this  kind,  any  two  of  them  will  work  truly  together,  pro- 
vided the  arcs  be  struck  through  the  point  t  ;  for  let  the 
wheel,  whose  radius  is  at,  be  removed,  and  another 
whose  radius  is  a' t  be  substituted,  a'p'  drawn  perpen- 
dicular to  T  p  p'  will  give  the  point  p'  as  the  centre  belong- 
ing to  the  arc  oy,  and  it  is  clear  that  this  new  arc  o  j  will 
work  as  well  with  kn  sls  the  former  one,  and  also  that  if 
the  radius  a't  had  been  substituted  for  bt  instead  of  for 
at,  by  placing  it  and  its  corresponding  line  a'p'  in  the 
situation  indicated  by  the  dotted  lines  b'q',  that  still  the 
conditions  of  the  construction  would  have  been  satisfied, 
and  those  two  wheels  worked  truly  together,  and  the  same 
may  Ih>  shown  of  any  other  pair  of  radii.  But  if  the  arcs 
wore  struck  through  a  point  wi,  not  coinciding  with  t,  then 
tho  wluH^ls  would  fall  into  two  groups,  in  one  of  which,  as 
at,  a't,  tho  arcs  are  struck  through  a  point  m  on  the  op- 
]H^ito  side  of  tho  line  of  centres  to  the  centre  points  p,  p', 
and  in  tho  other,  as  b  t,  b't,  they  are  struck  through  a 


APPEND.  A,]        ON   THE   TEETH    OF   WHEELS.  l65 

point  m  on  the  same  side  of  the  line  of  centres  as  the  cen- 
tre points  qq'.  Any  wheel  out  of  one  of  these  groups  will 
work  correctly  with  any  wheel  taken  from  the  other. 

But  suppose  that  a  pair  of  wheels  out  of  one  of  these 
groups  he  put  together,  for  example,  out  of  that  in  which 
the  point  m  and  the  centre  point  of  the  arc,  lie  on  opposite 
sides  of  the  line  of  centres,  and  let  a  t  and  b  t  he  the  radii 
of  the  wheels  in  question.  Now  the  relative  positions  of 
the  points  p  and  q  will  still  he  true,  but  the  arcs  will  no 
longer  be  struck  through  a  common  point,  one  of  them 
being  through  w,  the  other  through  mf  at  the  same  dis- 
tance on  the  opposite  side  of  x,  and  therefore  they  will  not 
work  truly  together.  The  arcs  of  the  entire  set  must 
therefore  be  struck  through  x,  and  then  any  two  wheels  of 
the  set  will  work. 

The  distance  xp  is  equal  to  ax  x  cos  axp,  and  if  axp 
be  fixed  at  75*"  3(y,  which  is  a  convenient  value,  then  xp  = 

— ,  whence  the  value  is  very  easily  found  for  any  given  ra- 
4 

dius,  for  in  this  case  the  value  depends  upon  the  rac^us 
alone  and  not  on  the  pitch  or  number  of  teeth,  as  in  the 
next  example.  The  practical  mode  of  setting  out  the 
teeth  has  been  already  explained. 

On  this  system,  however,  the  tooth  has  but  one  true 
point,  that  is  to  say,  it  is  only  strictly  exact  at  the  moment 
of  passing  the  line  of  centres,  and  I  therefore  greatly  pre- 
fer the  construction  about  to  be  described,  in  which  the 
side  of  the  tooth  is  made  up  of  two  arcs  united,  and  con- 
sequently has  two  points  of  accuracy.  The  tooth  just  de- 
scribed has  considerable  analogy  to  the  involute,  and  like 
it  has  the  fault  of  acting  with  too  great  a  degree  of  obli- 
quity. The  teeth  next  to  be  described  are  of  nearly  the 
same  form  as  that  which  has  been  so  long  in  use,  and  have, 
as  well  as  those  of  Fig.  8,  the  property  of  allowing  any 
pair  of  wheels  in  a  set  to  work  together. 


166  ON   THE   TEETH    OF   WHEELS.         f  APPEND.  A. 

TO   DESCRIBE   TEETH   CONSISTING  OF   TWO  ARCS  OP  CIBCLBS. 

Figures  9  and  10,  Plate  SO,  represent  a  pair  of  so  con- 
stituted teeth  in  contact,  fig.  9  shewing  their  action  before 
they  reach  the  line  of  centres,  and  Fig.  10  after  they  have 
passed  that  line ;  each  tooth  is  formed  of  two  arcs  of  cir- 
cles, a  J,  bcy  de^  ef^  of  which  the  concave  ones,  a&,  e/J  are 
situated  within  the  pitch  circles,  and  the  convex  ones,  ic, 
dcy  extend  beyond  these  circles;  therefore,  from  well  known 
principles,  the  concave  arc  a  h  will  drive  the  convex  arc  it^ 
until  the  point  of  contact  reaches  the  line  of  centres,  and  then 
the  convex  arc  h  c  will  begin  to  drive  the  concave  arc  ef. 

There  are  two  points  in  the  action  of  these  teeth  at  whidi 
perfect  accuracy  is  attained ;  one  of  them  is  when  the  teeth 
are  in  the  position  of  Fig.  99  during  the  mutual  action  of 
ab  and  de^  and  the  other  when  they  are  in  the  position  of 
Fig.  10,  during  the  action  of  be  and  ef;  and  the  arcs  are 
so  set  out  that  these  points  of  the  action  shall  take  place, 
the  one  nearly  in  the  middle  of  the  arc  of  motion  before  the 
line  of  centres  is  reached,  and  the  other  somewhere  about 
the  middle  of  the  arc  of  motion  that  is  traversed  from  the 
line  of  centres  until  the  teeth  quit  contact. 

The  construction  of  these  teeth  in  a  set  is  as  follows. 
AB,  Figures  9  and  10,  is  the  general  direction  of  the  Kne 
of  centres;  qpt,  as  before,  is  a  line  making  a  constant 
angle  of  75"*  with  the  line  of  centres ;  k  t  k  perpendicular 
to  QPT  and  having  its  two  points  k  set  off  at  equal  distances 
on  each  side  of  t,  these  points  and  the  lines  being  inva- 
riable for  the  entire  set. 

The  centres  for  the  convex  arcs  are  found  by  joining  the 
centre  of  each  wheel  (a.  Fig,  10 ;  b.  Fig.  9)  with  that 
point  K  which  lies  on  the  opposite  side  of  the  line  qpt. 
Thus  in  Fig.  9,  Q  is  the  centre  of  the  convex  arc  de,  found 
by  joining  bk,  and  in  Fig.  10,  p  is  the  centre  of  the  convex 
arc  bCf  found  by  joining  ak. 


APPEND.  A.]    ON  THE  TEETH  OF  WHEELS.  l67 

The  centres  for  the  concave  arcs  are  found  by  joining 
the  centre  of  each  wheel  with  the  k  which  lies  between 
it  and  the  line  qpt  ;  thus  in  Fig.  9*  p  is  the  centre 
of  the  concave  arc  ahj  found  by  joining  ak,  and  pro- 
ducing it  to  meet  pqt,  and  in  Fig.  10,  q  is  the  centre  of 
the  concave  arc  efi  found  by  joining  bk,  and  producing  it 
to  meet  tpq.  Moreover,  the  whole  of  these  concave  and 
convex  arcs  are  struck  through  a  point  lying  beyond  t  at  a 
constant  distance,  Tn,  or  Tm,  which  for  simplicity's  sake  I 
have  assumed  equal  to  half  the  pitch ;  finding  that  this 
will  place  the  correct  points  of  the  action  at  a  sufficient  dis- 
tance o  each  side  of  the  line  of  centres. 

The  consequences  of  this  arrangement  will  be,  that  any 
pair  of  teeth  so  described  will,  when  put  together,  answer 
the  conditions  of  the  construction  already  demonstrated. 

(Fig.  6.) 

Ist  (Fig.  90  Before  reaching  the  line  of  centres  we 
have  a  concave  arc  ah  driving  a  convex  one  de,  of  which 
the  first  has  been  struck  from  a  centre  p,  derived  from  its 
nearest  k,  and  the  second  from  a  centre  q,  derived  from  its 
farthest  k,  consequently  both  derived  from  the  same  k  ; 
also  the  arcs  have  both  been  struck  through  a  point  m,  at 
the  same  distance  beyond  t,  and  therefore  will  work  truly 
together. 

Sd.  (Fig.  10.)  After  passing  the  line  of  centres,  a 
coiiYex  arc  be  drives  a  concave  arc  efj  which  in  like  man- 
ner are  seen  to  have  been  derived  from  the  same  k,  and  to 
have  been  struck  through  a  common  point  /i,  so  that  al- 
though the  position  of  all  these  points  is  reversed,  the  arcs 
will,  in  this  case,  work  truly  together. 

The  same  will  manifestly  be  true  for  every  pair  of  wheels 
in  the  set,  for  the  distances  tk  and  Tm,  or  tti,  are  con- 
stant for  the  whole. 


168  ON   THE   TEETH   OF   WHEELS.         [ APPEND.  A. 


CONSTRUCTION  OF   THE  ODONTAORAPH. 


To  enable  a  workman  to  find  these  points  p  and  q  at 
once  in  every  case,  I  have  contrived  the  instrument  ah^y 
described,  (vide  page  150,)  which  I  have  termed  an  Odon- 
tagraph,  and  have  represented  in  Fig.  11,  Plate  20,  with 
the  arrangements  for  describing  the  tooth  fed  of  Figures 
9  and  10.  These  three  drawings  being  all  made  to  the 
same  scale  will  explain  each  other  by  comparison. 

The  instrument,  as  already  mentioned,  consists  of  a  kind 
of  bevil  formed  of  a  sheet  of  card  paper,  four  times  the 
lineal  size  of  the  drawing  ef^d,  the  angle  Btkia  75%  and 
the  side  A:^f  is  occupied  by  a  scale  of  equal  parts  numbered 
from  t  both  ways.  An  example  will  show  how  this  instra- 
ment  is  connected  with  the  previous  demonstration. 

Let  the  example  be  a  wheel  of  26  teeth  4  inch  pitch. 
Describe  an  arc  ter  of  the  required  pitch  circle,  and  set 
off  upon  it  ^T  equal  to  the  pitch  and  bisected  in  ^,  draw  ra- 
dial lines  Bty  B  T.  To  describe  the  arc  ef  within  the  pitch 
circle,  apply  the  slant  edge  d  <  of  the  scale  to  the  upper 
radial  line  b<,  as  in  the  figure.  In  the  table  headed  "  Cen- 
tres  for  teeth  within  the  pitch  circle,"  look  down  the 
column  of  2  inch  pitch,  and  opposite  to  26  teeth  will  be 
found  the  number  37,  which  being  doubled  gives  74-  The 
point  indicated  on  the  drawing  board  by  the  position  oT 
this  number  at  q  on  the  scale  of  equal  parts  <f,  which, 
is  marked  Scale  of  centres  for  teeth  within  pitch  circle^. 
in  the  actual  instrument,  is  the  centre  required,  fronm 
which  the  BTcfe  must  be  drawn  with  a  radius  qe. 

Now  a  comparison  of  this  figure  with  Fig.  10,  will  sho\iir 
that  thus  far  the  relative  positions,  inclinations  and  dis- 
tances  have  been  indicated  by  the  instrument  for  the  point 
q  (the  Q  of  Fig.  10.)     The  line  b<.  Fig.  11,  is  the  same  as 
BT  in  Fig.  10. 

The  centre  for  the  arc  erf,  which  lies  outside  the  pitch 


>•] 


ON    THE    TEETH    OF    WHEELS. 


169 


circle,  is  found  in  a  manner  precisely  similar,  by  applying 
the  slant  edge  of  the  scale  to  the  lower  radial  line  bt, 
placing  the  instrument  in  the  position  indicated  by  the 
(lotted  lines.  The  table  of  centres  for  t«eth  outside  the 
foleh  circle  does  not  contain  SG  in  its  column  of  Number 
of  Teeth,  therefore  the  nearest  number  must  be  taken, 
which  in  this  case  is  30,  and  the  number  28  =  ^  x  14 
in  the  column  of  2  inch  pitch,  will  indicate  the  position  of 
the  centre  q  upon  the  scale  t  A  of  centres  for  teeth,  outside 
Hupitch  circle,  this  scale  being  so  titled  in  the  actual  instru- 
Here,  again,  a  comparison  of  Fig.  11  with  Fig.  9. 
ill  show  that  this  new  operation  has  given  the  true  relative 
wdon  of  the  jmint  q  to  the  radial  line  bt  and  arc  de. 
]  will  now  explain  in  a  few  words  the  mode  of  calcidating 
le  numbers  in  the  table,  by  way  of  enabling  other  persons 
I  liter  any  of  the  conditions.  A  formula  tor  tliese  num- 
a  may  be  obtained  as  follows.  (Vide  Fig.  6, 'page  l62.) 
wn  A  draw  am  perpendicular  to  ttp',  then  from  the 

lailar  triangles  amp,  ptk  we  obtain  kt  = ^         ~ 


Let  KT  =c,  at  =  r,  pt  = 
D.R.  sin  0 


(1) 


R,  COS  0  —  d' 

^'Now  the  point  p  being  in  this  case  obtained  from  the  k 

the  opposite  side  of  t  to  a,  this  formula  belongs  to  that 

rt  of  the  tooth  which  lies  beyond  the  pitch  circle,  accord- 

[to  the  principles  already  laid  down.     If  tk"  be  taken 

lal  to  T  K  on  the  line  k  t  produced,  and  a  point  p"  ob- 

Kd  by  joining  ak",   and  producing  the  line  to  meet 

'T,  then  p"  will  belong  to  the  part  of  the  tooth  within 

pitch  circle,  and  the  similar  triangles  amp",  p"tk", 

igive  us  for  this  case  the  formula 

d'h  sin  fl  ,  ^        ,  ,  „ 

C= ;; i (2);  where  d=tp'. 

B  cos  tf  +  D  ^  -^ 

pw  the  value  of  c,  which  represents  the  equal  lines 


170  ON   THE   TEETH    OF   WHEELS.       [aPPEND«  A. 

KT,  or  K^^T,  may  be  determined  for  the  whole  set,  by  con- 
siderations similar  to  those  already  employed  in  settling 
the  diameter  of  the  constant  describing  circle  in  the  first 
section  of  this  paper.  If  the  radius  at  of  a  wheel  be  as- 
sumed of  such  a  length  that  a  k^'  fall  perpendicularly  upon 
k''t,  then  will  the  line  ak'^p''  become  parallel  to  ptp'', 
and  consequently  the  point  ^"  will  go  off  to  infinityi  and 
the  arc  which  should  be  struck  from  it  to  form  the  flank  (tf 
the  tooth  will  become  a  right  line  perpendicular  to  ptp". 

If  the  radius  at  be  taken  still  smaller  with  respect  to 
k^'t,  it  will  be  seen  (by  taking  k't  larger  than  at)  that  in 
such  a  case  the  point  p,  will  make  its  appearance  on  the  qp* 
posite  side  of  t  *,  but  this  makes  the  flank  of  the  tooth 
convex,  and  drawing  inwards  so  as  to  be  less  at  the  base 
than  at  the  pitch  line,  which  is  an  impracticable  form.  To 
avoid  this,  and  at  the  same  time  to  make  k^^t  as  large  as 
possible  consistently  with  this  limitation,  assume  k^^t  equal 
to  R^  sin  6 J  where  r^  is  the  least  radius  of  the  set.    This 
value  corresponds  to  the  case  in  which  a  yl"  is  perpendico- 
lar  to  k'^t,  and  necessarily   excludes   the    impracticable 
forms ;  for  since  the  least  radius  of  the  set  now  corresponds 
to  that  peculiar  example  in  which  ak'^p'^  is  parallel  to  ptp", 
every  other  value  of  at  being  larger,  will  throw  the  points 
v^'  on  the  opposite  side  of  t  to  m,  which  is  the  thing  re- 
quired to  produce  the  concave  flank.     These  observations 
apply  only  to  that  value  of  kt  which  lies  nearest  the  centre 
A,  and  therefore  to  the  flank  or  portion  of  tooth  within  the 
pitch  circle.     As  to  the  opposite  value  of  t  k,  which  cor- 
responds to  the  portion  of  tooth  beyond  the  pitch  circle, 
and  which  it  must  be  remembered  is  equal  to  tk'^  it  is 
clear  from  the  figure  that  whatever  value  be  given  to  it,  its 
point  p  will  always  lie  between  t  and  m,  and  the  arc  of 
tooth  be  convex,  supposing  it  to  be  struck,  as  it  must  boi 
through  a  point  near  to  t. 

*  Our  fonnula  then  becomes  c  z:  — ^ . 

d'  —  B  cos  6 


The  ^-alue  selected  for  k"t  (namely  r'  sin  6)  will  tbere- 
isaitKT.     Substitute  now  this  value  for  c  in  the  for- 

B  (1)  and  ('2),  and  after  arranging  the  terms  we  obtain 

following  values  of  d  and  d'. 


'•] 


ON    THE    TEETH    OF    WHEELS. 


171 


h'b  cos  S 


.  .  (3)       andD'=- 


•  C-t) 


low  D  and  u'  (that  is  tp  and  tp")  are  the  distances  of 
centre  points  of  the  arcs  measured  from  t,  and  it  will 
leen  by  comparing  the  diagrams  with  the  description  of 
Odontagraph,  that  the  numbers  in  the  columns  of  each 
ii  are  the  values  of  d  and  d',  corresponding  to  the  num- 

of  teeth  in  each  wheel  given  in  the  first  column,  or, 

1  is  the  same  thing,  to  the  values  of  the  radii  n  and  it', 
find  these  numbers  for  a  given  pitch,  substitute  in  (3) 
(♦)  the  particular  values  of  r'  and  B,  and  by  help  of 
ble  of  logarithms,  the  values  of  u  and  d'  belonging  to 

my  values  of  r  as  may  be  thought  necessary,  may  be 
puted,  and  thus  the  column  of  numbers  obtained  for 
;  pitch.  Tlioae  of  the  other  pitches  may  be  derived 
\  the  first  by  common  proportion.  In  this  way  I  formed 
table,  assuming  12  tor  the  least  number  of  teeth,  and 
for  the  value  of  8,  and  employing  a  scale  of  half  inches 
tenths  in  which  to  express  the  values  of  d  in  the  near- 
irbole  numbers,  because  I  foimd  that  a  unit  of  the  twen- 
b  of  an  inch  was  sufficiently  small  to  avoid  practical  error. 
t  i»  unnecessarj'  to  have  numbers  corresponding  to 
y  wheel,  for  the  error  produced  by  taking  those  which 
Dg  to  the  nearest  as  directed,  is  so  small  as  to  be  un- 
wiable  in  practice.  I  have  calculated  the  amount  and 
re  of  these  errors  by  way  of  obtaining  a  principle  for  the 
ber  and  arrangement  of  the  wheels  selected.  It  is 
Bcessarj-  to  go  at  length  into  these  calculations,  which 

,  &om  very  simple  considerations,  but  I  will  briefly 
I  the  results. 

e  difference  of  form  between  the  tooth  of  one  wheel  and 
Other  ia  due  to  two  causes,  (1)  the  difference  of  curva- 


1J2  ON    THE   TEETH    OF   WHEELS.        FaPFEND.  A. 


ture,  which  is  provided  for  in  the  Odontagraph  by  placing  the 
compasses  at  the  different  points  of  the  scale  of  equal  parts, 
(2)  the  variation  of  the  angle  /bt,  (Fig.  11,)  which  is  met 
by  placing  the  instrument  upon  the  two  radii  in  succession. 

The  first  cause  is  the  onlv  one  with  which  these  calcukr 
tions  are  concerned.  Now  in  three  inch  pitch  the  great- 
est difference  of  form  produced  by  mere  curvature  in  the 
portion  of  tooth  which  lies  beyond  the  pitch  circle,  is  only 
•Qt  inch  between  the  extreme  cases  of  a  pinion  of  twelve 
and  a  rack,  and  in  the  acting  part  of  the  arc  within  the 
pitch  circle  is  *1  inch,  so  that  as  all  the  other  forms  lie  be- 
tween these,  it  is  clear  that  if  we  select  only  four  or  five 
examples  for  the  outer  side  of  the  tooth  and  ten  or  twelve 
for  the  inner  side,  that  we  can  never  incur  an  error  of  more 
than  the  to  oth  of  an  inch  in  three  inch  pitch  by  always 
taking  the  nearest  number  in  the  manner  directed,  and  a 
proportionably  smaller  error  in  smaller  pitches.  But  to 
ensure  this,  the  selected  numbers  should  be  so  taken,  that 
their  respective  forms  shall  lie  between  the  extremes  at 
equal  distances.  Now  it  appears  that  the  variation  of  finrm 
is  much  greater  among  the  teeth  of  small  numbers  than 
among  the  larger  ones,  and  that  in  fact  the  numbers  in  the 
two  following  series  are  so  arranged  that  the  curves  cor- 
responding to  them  possess  this  required  property. 

For  the  outer  side  of  the  tooth,  12,  14,  17,  21,  26^  34^ 
47,  73,  148,  Rack. 

For  the  inner  side,  12,  13,  14,  15,  16,  17,  19,  22,  26, 
33,  46,  87,  Rack. 

Now  these  numbers,  although  strictly  correct,  would  be 
verj'  inconvenient  and  uncouth  in  practice  if  employed  for 
a  table  like  that  in  question,  where  convenience  manifegtly 
requires  that  the  numbers,  if  not  consecutive,  should  always 
proceed  either  by  twos  or  fives,  or  by  whole  tens,  and  so  on. 
They  arc  only  given  as  guides .  in  the  selection,  and  bj 
comparing  them  with  the  actual  table,  their  liae  in  flie 
formation  of  the  first  column  will  be  evident 


ESSAY     II. 

ON    THE    SHAFTS   OF    MILLS. 


CHAPTER  I. 

IHl.  To  make  these  Essays  useful  to  operative   mecha- 

Inica;  to  save  engineers  and  managers  of  manufactories 
ibe  trouble  of  much  explanation  in  giving  directions  to 
fcremen  and  others,  who  are  to  carry  their  ideas  into  effect ; 
lo  give  workmen  some  notion  of  the  principles  on  which 
their  work  should  be  conducted ;  and  to  construct  machi- 
nerj  upon  true  principles,  which  is  ultimately  the  most 
economical  plan  of  proceeding ;  we  introduce  the  following 
extract  from  a  respectable  periodical  publication*,  as  it 
appears  appUcable  to  the  subjects  of  these  papers,  and  may 
induce  the  reader's  taste  for  entering  upon  a  new  and  un- 
trodden path. 

"  A  country  in  which  manufactures  are  extensively 
fslablished,  and  conducted  with  spirit,  as  in  Britain,  be- 
'■omes  by  degrees  a  country  of  machinerj-.  For  invon- 
ti'ins  to  diminish  the  quantity  of  human  labour  employed, 
"ill  be  more  ingenious  in  construction,  more  powerful  in 
opi-ration,  and  of  more  general  use,  in  proportion  to  the 
iitessity  of  furnishing  a  greater  quantity  of  commodities 
^'  moderate  and  equable  prices.    The  bodily  exertions  of 

•  Kt-lectic  RiivJew,  Dec.  180C,  Art.  XII. 


174  ON    THE    SHAFTS   OF    MILLS.  [ CHAP.  I. 

workmen,  in  whatever  branch  of  labour,  have  their  limits, 
and  excessive  efforts,  if  unduly  prolonged,  irremediably 
destroy  the  health  and  vigour  of  those  who  pursue  thenL 
But  machines  may  be  continued  in  activity  day  and  night, 
week  after  week,  and  month  after  month ;  having  in  them- 
selves no  life  which  suffers  a  sensible  consumption,  no  prin- 
ciple of  activity  whose  energy  requires  a  pause  to  effect  its 
recovery  or  renovation. 

"  We  have  seen  the  manufactures  of  our  own  country 
solicit  the  aid  of  every  hand  that  could  be  spared  from  its 
agriculture,  and  seek  in  distant  lands  for  labourers  of 
every  age  to  supply  the  mill  or  to  throw  the  shuttle.  We 
have  seen  ingenuity  exerted  to  its  utmost,  to  contrive  and 
to  construct  those  machines  which  these  labourers  were  to 
superintend  and  assist.  We  remember  the  time  when 
these  constructions  were  the  dread  and  the  hatred  of  the 
manufacturers,  but  we  believe  the  most  ignorant  workman 
of  the  present  day  acknowledges  their  utility,  and  would 
with  difficulty  be  induced  to  relinquish  that  very  imple- 
ment which  his  father  or  grandfather  would  have  gladly 
committed  to  the  flames. 

"  Considering  then  the  importance  of  machines  to 
shorten  labour,  and  the  number  of  persons  who  are  mter- 
ested  in  them,  as  proprietors,  as  inventors,  or  as  con- 
structors, it  is  wonderful  that  so  little  has  hitherto  been  com- 
municated on  this  subject  by  the  medium  of  the  press. 

"  The  process  towards  perfection  in  complicated  machi- 
nery is  perhaps  too  generally  the  reverse  of  what  might  be 
expected.  When  practice  has  shewn  the  importance  of  a 
machine,  science  takes  it  up,  investigates  its  principles, 
analyses  its  movements,  and  connects  them  by  the  assist- 
ance of  mathematical  precision. 

^*  Mathematicians  are  seldom  inventors,  and  workmen 
are  rarely  men  of  science,  yet  the  mutual  assistance  of 
study  and  practice  is  necessary,  to  perfect  the  subject  which 


■0 


ON    THE    SHAFTS    OF    MILLS, 


175 


each  is  intent  on  improving."  It  was  Buchanan's  aim, 
then,  to  come  between  these  two  classes,  and  to  make  them 
better  acquainted,  and  more  useful  to  each  other.  How 
far  he  has  succeeded  in  the  attempt,  must  be  left  to  the  de- 
termination of  time. 

Lin  common  with  all  writers  on  similar  subjects,  he  cx- 
prienced  considerable  difficulty  in  finding  precise  technical 
mirds  to  express  the  different  parts  of  mill-work.  Those 
wiiich  are  used  by  millwrights  in  different  districts  being 
Teiy  different  from  each  other.  It  is  hoped,  however,  that 
the  explanations  which  we  have  given  of  the  terms,  will 
niake  them  sufficiently  clear. 

LWith  regard  to  this  particular  Essay  on  the  Shafts  of 

I,  the  subject  is  treated  in  a  manner  similar  to  that 

wed  in  the  Inquiry  into  the  Strength  and  Durability 

of  (he  Teeth  of  Wheels.     For  the  reasons  there  given, 

Buchanan  did  not  here  enter  into  the  demonstrations  of 

the  elementar)'  propositions  which  serve  to  guide  the  in- 

qniry.     He  was  at  pains,  however,  to  collect  and  arrange 

etB  respecting  gudgeons  and  journals  in  actual  use,  upon 

)kh  to  ground  calculations.     This  method  he  considered 

being  much    more    certain    than  rearing   calculations 

I  insulated  experiments  made  on  the  cohesive  strength 

.materiais.     These,  however,  are  of  great  value,  and 

tome  cAses  he  has  endeavoured  to  apply  them.     The 

ious  Tables  given  in  the  course  of  the  Essay,  will  be 

peat  use  to  the  millwright  in  finding  without  trouble 

lizes  of  gudgeons  and  journals  for  any  case  that  may 

r  in  practice,  and  the  principles  on  which  they  are 

fd  are  laid  down  in  so  plain  a  manner  that  he  will 

iy  understand  and  apply  them.     These  Tables  may  be 

Indered  as  certain  great  lines  drawn  to  guide  the  mill- 

[ht  in  his  work,  and  even  allowing  they  may  not  be  ab- 

tely  true,  he  may  find  from  experience  how  near  they 

^  be  approached  with  safety. 


176  ON   THE   SHAFTS   OF   MILLS.  [CHAP.  I. 


18&  To  proportion  the  diameters  of  axles  to  the  stress 
they  have  to  hear,  is  in  mill-work  of  great  practical  import- 
ance. On  the  one  hand,  if  the  shafts  he  made  too  weak, 
it  is  evident  they  must  soon  give  way ;  and  on  the  oth^ 
hand,  if  made  too  strong,  they  occasion  not  only  unnecessary 
expense  in  the  construction  of  the  machinery,  hut,  what  is  in 
most  cases  still  worse,  a  waste  of  power  from  unnecessary 
friction.  It  is  therefore  desirable,  that  the  millwright 
should  have  some  rules  to  guide  him  in  this  very  important 
part  of  his  business ;  a  part  which  has  hitherto  in  most 
cases  been  conducted  entirely  at  random.  This  Essay  gives 
such  a  practical  view  of  the  subject  as  shall  enable  the  mill- 
wright to  proceed  with  greater  certainty. 

183.  Until  of  late  years,  most  of  the  shafts  used  in  mill- 
work  were  constructed  of  timber.  The  use  of  cast  iron  in 
this  and  other  parts  of  mill- work,  however,  has  now  become 
almost  universal.  For  this  improvement  we  are  perhaps 
indebted  to  those  who  are  engaged  in  the  cotton  manufiic- 
ture.  After  Arkwright's  invention,  it  became  a  great  ob- 
ject with  them  to  save  time  in  the  erection  of  machineiy, 
and  to  render  it  as  durable  as  possible ;  for  every  stoppage 
was  attended  with  great  loss,  by  throwing  idle  the  numbers 
of  people  necessary  in  cotton  mills. 

Besides  the  expense  attending  the  repair,  what  had  per- 
haps still  more  weight  with  them  was,  that  the  profits  at 
that  period  on  cotton  spinning,  were  almost  unparalleled  in 
any  other  branch  of  manufacture. 

Another  circumstance  which  tended  very  much  to  the 
advancement  of  mill-work,  arose  from  James  Watt's  im- 
provement of  the  steam  engine,  which  enabled  cotton-spin- 
ners and  other  manufacturers  who  required  power  to  work 
their  machinery  to  carry  on  their  business  in  towns.  Hence 
power  and  people  might,  without  trouble,  be  concentrated 
on  the  most  eligible  spot,  and  the  great  expense  and  disad- 
vantages avoided  which  are  attendant  on  colonizing  the 


CHAP.  1-3  ON   THE   SHAFTS    OF   MILLS.  177 

remote  situatioiis  in  which  powerful  fells  of  water  are  com- 
monly found.  The  questions  of  health  and  morals  belong 
to  the  l^islator,  not  to  the  civil  engineer. 

The  introduction  of  cast  iron  then  may  be  considered  as 
a  kind  of  new  era  in  the  history  of  mills,  without  the  use  of 
iMch  it  would  not  have  been  possible,  with  the  same  num- 
ber of  operative  mechanics,  to  have  constructed  one  tenth 
part  of  the  machinery  which  has  of  late  years  been  erected 
in  Great  Britain. 


CHAPTER   11. 


SECTION  I. 


GENERAL   DESCRIPTION   OP  8HAPT8. 


184.  The  axles  used  in  mill-work  are  commonly  den< 
minated,  when  of  a  large  size,  sfu^ ;   those  which 
smaller,  are  usually  called  spindles.     Thus,  for  exampl 
we  say  the  shaft  of  a  water-wheel ;  the  spindle  which 
ries  the  millstone  of  a  com-milL 

185.  When  shafts  lie  in  a  horizontal  direction,  they  are 
called  It/ing  or  horizontal  shafts ;  when  vertical,  they  axe 
termed  upright  or  vertical  shafts. 

186.  Shafts  are  usually  made  of  wood  or  of  iron.     Large 
wooden  shafts  are  generally  made  either  of  solid  oak,  or  are 
built  of  fir-logs.     The  scarcity  of  large  oak  occasioned  the 
built  shafts  of  fir  to  come  into  more  general  use.     The 
latest  improvement  made  on  wooden  shafts,  was  that  of 
having  what  are  called  cross-tailed  gudgeons  * .    Before  that 
improvement,  it  was  attended  with  very  great  trouble  and 
expense  to  keep  the  gudgeons  from  becoming  loose  in  the 
shafts.     Indeed,  it  was  found  impracticable  to  keep  them 
fast  for  any  considerable  time. 

187.  Fig-  1>  Plate  XL  represents  a  wooden  shaft,  with 
the  gudgeons  in  use  previously  to  the  last  improvement. 
They  are  called  laid-in  gudgeons,  a  b  c  is  the  gudgeon 
somewhat  in  the  form  of  the  letter  T.  One  of  the  tails,  c, 
was  let  into  a  mortice,  and  the  rest  of  the  gudgeon  sunk  into 

*  The  gudgeon  is  the  arbour  or  spindle  on  which  the  shaft  turns. 


CHAP.  II.]  ON    THi*-.^^  .^S    OF    MILLS.  179 

its  place  in  the  centre  of  the  shaft.  In  order  to  accomplish 
this,  it  was  necessary  to  cut  out  the  part,  from  b  to  d, 
Fig.  1,  No.  2.  After  the  gudgeon  was  laid  in  its  place,  the 
vacant  part  was  filled  up  by  the  piece  of  wood  d  e,  Fig.  1, 
No.  1.  The  shaft  was  then  hooped,  and  the  end  of  it 
driven  full  of  wedges,  in  order  to  fasten  the  gudgeon.  This 
gudgeon  is  shown  in  perspective.  Fig.  1,  No.  3. 

188.  Fig.  2.  represents  a  wooden  shaft,  with  cross- tailed 
gudgeons.  This  kind  of  gudgeon  is  made  of  cast  iron,  and 
being  thin  in  the  cross-tails,  let  in  from  the  end  of  the  shaft, 
it  leaves  the  wood  much  more  entire  than  the  laicUin  gud- 
geotij  while  its  cross-arms  take  a  much  firmer  hold.  After 
it  is  let  in,  the  hoops  are  driven  on  the  end  of  the  shaft, 
when  warm,  and  lay  firm  hold  of  the  ends  of  the  cross-tails. 
The  wood  is  then  wedged  up,  which  makes  the  gudgeons 
perfectly  fast.  Fig.  3.  is  a  perspective  view  of  a  cross-tailed 
gudgeon,  and  Fig.  4.  its  profile.  It  is  cast  with  the  round 
part  undermost ;  for  which  reason  the  pattern  must  have  a 
taper,  to  make  it  rise  out  of  the  sand.  This  taper  has,  in 
the  cross-tails,  another  very  important  use,  that  of  giving 
the  gudgeon  the  advantage  of  dove-tailed  joints  with  the 
timber  of  the  shaft  when  it  is  wedged  up.  Instead  of 
wrought  iron  hoops,  cross-tailed  gudgeons  sometimes  have 
a  hoop  of  cast  iron  cast  along  with  the  tails,  as  represented 
by  Fig.  5. 

189*  When  it  is  considered  that  the  direction  of  the 
stress  which  tends  to  loosen  the  gudgeon  in  a  wooden  shaft 
is  continually  changing,  and  that  such  action  is  exerted 
upon  wood,  a  material  that  is  so  very  easily  permanently 
compressed,  it  will  not  be  wonderful  that  it  should  have 
been  found  difficult  to  render  them  firm  and  lasting.  The 
last  method,  viz.  that  where  the  hoop  is  cast  along  with 
the  cross-tails,  seems  to  be  far  preferable  to  the  other ;  but 
perhaps  it  seldom  happens  that  the  hoop  part  is  of  sufficient 
length.     It  may  be  proved,  that  when  the  diameter  of  the 


180  ON   THE   SHAFTS   OF   MILLS.  [CHAF.  XL 

shaft  is  sufficient  for  the  stram  upon  it,  the  length  of  tb6 
hoop  should  he  equal  to  the  square  of  the  diameter  pf 
the  shaft  divided  hy  the  length  of  the  shaft;  otherwise 
there  will  be  no  certainty  of  the  gudgeon  remauung 
permanently  fixed* 

190.  An  improved  method  of  fixing  gudgeons  is  do- 
scribed  in  the  Transactions  of  the  Society  of  Arts,  &c.f 
Vol.  xxxi.  p.  223 ;  it  consists  in  casting  the  gudgeon  with 
cross-arms,  which  fit  into  proper  notches  in  an  octagonal 
box  of  cast  iron  that  has  been  previously  fixed  upon  the 
end  of  the  shaft.  The  arms  of  the  gudgeon  are  retained 
in  their  places  by  screw-bolts. 

In  Plate  [IV.  A,]  Fig.  1,  2,  and  3,  a  a  represents  the 
end  of  the  wooden  shaft,  which  is  supposed  to  be  made  of 
an  octagonal  form,     b  b  is  the  cast  iron  box  accurately  fit- 
ted on  the  end  of  the  shaft,  and  wedged  tight.     The  end  of 
the  box  has  a  projecting  flanch  a  a,  with  four  notches  to  re- 
ceive  the  cross-arms  bbjdd  of  the  gudgeon  c.    These  cross- 
arms  are  firmly  fixed  to  the  box  by  four  screw-bolts,  which 
pass  through  the  flanch,  and  the  ends  of  the  cross-arms. 
The  section.  Fig.  3,  shews  the  box  b  b  on  the  end  of  the 
shaft,  with  the  gudgeon  c,  and  its  cross-arms  separated ;  to 
explain  a  further  precaution  which  is  necessary  for  strength. 
This  precaution  consists  in  the  cross-arms  having  projec- 
tions, e  Cy  which  enter  the  end  of  the  box,  and  keep  the  gud- 
geon true  to  its  centre,  and  prevent  any  lateral  strain  on 
the  bolts.     When  the  gudgeon  of  a  wheel  is  fitted  accord- 
ing to  this  method,  it  can  be  easily  removed  when  it  is  so 
far  worn  that  a  new  one  is  necessary ;  and  the  new  one  may 
be  inserted  without  injury  to  the  end  of  the  shaft. 

This  improved  method  was  invented  by  Robert  Hughes. 

It  is  obvious  that  the  length  of  the  box  should  be  regu- 
lated by  the  rule  stated  in  Art.  189*.  The  real  advantage 
gained  by  this  mode  of  fixing  seems  to  be,  that  of  retaining 

*  The  rule  suppotjes  the  shaft  to  be  proportioned  to  the  stress  upou  it 


ICH.1!'.  It.]  ON   THE   SHAFTS    OF    MILLS.  181 

f  the  end  of  the  wooden  shaft  more  perfect,  with  the  means 
\(d  renewing  the  gudgeon,  without  injurj*  to  the  shaft, 
lyi.  Cast  iron  shafts  are  sometimes  made  hollow  cylin- 
ders, and  sometimes  they  are  made  solid,  and  of  various 
figures.     It  is  demonstrable,  that  a  hollow  cylinder  is  much 
stronger,  with  the  same  quantity  of  matter,  than  it  woidd 
hfl  if  made  into  a  solid  of  the  same  length.     This  law  is 
YCij  observable  in  the  beautiful  economy  of  nature ;  for  in- 
slance,  the  stalks  of  plants,  the  quills  of  birds,  the  bones 
(^animals.     But,  in  the  works  of  art,  numberless  obstacles 
■to  perfection  continually  occur.     In  this  particular  case  the 
■ttpcnse  of  making  small  shafts  hollow,  would  be  very  great  j 
(1  another  objection  is,  the  difficulty  of  making  such  cast- 
s  perfect.     .Shafts  of  a  small  diameter  are,  therefore, 
monly  made  solid. 
,  Fig.  6,  Plate  II.  represents  a  cast  iron  cylindrical 
It  consists  of  three  parts,  the  body,  a  u  c  d,  and  the 

0  gudgeons,  a  e  c,  and  B  i'  D  c.    The  gudgeons  are  turned 
d  carefully  fitted  into  the  ends  of  the  body,  which  is  bored 

1  turned  to  receive  them.  They  are  then  fixed  with 
sorew-bolts,  which  pass  through  the  flanches,  as  may  be  seen 
!j_v  the  figure  •- 

This  kind  of  shaft  will  obviously  be  variously  constructed, 
according  to  circumstances.  That  represented  in  the 
figure  was  made  for  a  cast  iron  water-wheel.  The  use  of  the 
small  projections  h,  h,  &c.,  is  to  prevent  the  eye  of  the  arms 
from  shifting  roxmd  on  the  shaft.  When  cylindrical  shafts 
are  not  used,  what  are  called  feathered  shafts  are  often 
I    adopted. 

^Kl93.  Fig  7,  Plate  III.  represents  this  construction  of  a 
^Hiift.  It  probably  took  its  name  from  its  resemblance  to 
The  feathered  part  of  an  arrow.     It  may  be  here  remarked, 

In  this  constroction,  the  resUtoncc  to  twisting  depends  entirely  on  the 
difEcultiet  appear  to  be  encoimtereil  jn  costing  without  cor- 
indiiig  wlnuitngee,  cilhcr  in  strength  or  beauty. 


^rik,  and  eomc  d 
^^bmidiiig  wlnui 


18S  ON   THE    SHAFTS   OP  HILLS.  [CHAF.  H. 

that  shafts  of  this  species,  as  often  constructed,  are  by  do 
means  calculated  to  withstand  the  twist  brought  upon  them 
by  the  strain  of  the  machinery.  From  the  breadth  of  tlie 
feathers,  their  strength  to  withstand  lateral  pressure,  is,  no 
doubt,  considerable  -,  but  wanting  substance  between  the 
feathers,  they  are  liable  to  continual  tremor.  Where  fea- 
thers are  applied  to  shafts  it  is  preferable  to  keep  the  body 
of  the  shaft  fully  as  strong  as  the  gudgeon,  or  journal* ^  and 
apply  the  feathers  merely  to  prevent  bending  in  the  mid- 
dle, as  Fig.  7)  No.  4.  But  the  simple  square.  Fig.  8,  is 
more  easily  made,  and  has  been  found  in  practice,  at  least 
as  advantageous  as  any  other  form  that  has  been  tried  for 
solid  shafts  t. 

Having  given  this  general  account  of  shafts,  we  come 
next  to  consider  the  causes  from  which  the  stress  on  them 
arises. 

SECTION  11. 

OF  THB  KINDS  OF  STRESS  TO  WHICH   SHAFTS   ARE  SUBJECT. 

194.  There  are  two  kinds  of  stress  to  which  shafts  are 
liable :  first,  lateral  stress^  by  which  they  may  be  broken 
across:  secondly,  stress  arising  from  torsion^  by  which 
they  may  be  wrenched  or  twisted. 

All  horizontal  shafts  are   liable  to  the  first  kind  of 
stress,  viz.  lateral  stress ;   and  some  have  no  other  strain 

*  Journals^  or  journeys^  are  gudgeons  subject  to  torsion. 

t  It  is  easily  proved,  that  the  best  form  for  a  revolving  shaft  is  a  cylinder, 
and  tliat  in  any  other  form  the  flexure  will  be  irregular,  and  consequently 
produce  irregular  wear  on  the  gudgeons  and  brasses ;  but  when  a  shaft  is 
to  be  adapted  for  placing  wheels  on  any  part  of  its  length,  a  square  section 
is  convenient ;  in  all  other  cases,  the  section  ought  to  be  circular  with  pro- 
jections, as  at  H,  H,  Fig.  6,  Plate  II.  By  making  four  of  these  projections 
continue  throughout  the  length  upon  a  cylindrical  shaft,  all  the  advantage 
and  convenience  of  a  square  one  would  be  obtained,  with  very  little  irregular 
flexure.  The  projections  should  not  be  greater  than  is  necessary  to  ^x  the 
wheels  firmly  on  the  shaft. 


ICHAP.  11.]  ON    THE    SHAFTS    OF    MILLS. 

I  whatever ;  as,  for  instance,  a  water-wheel  shaft,  where  the 
motion  is  conuuunicated  from  teeth,  on  the  shrouding.  See 
iPlatelll.  Fig.  9. 

In  Fig.  10,  the  stress  on  the  upright  shaft  arises  from 
Itorsion  only ;  excepting  what  may  proceed  from  the  inac- 
Jturacy  of  the  teeth  of  the  wheels,  which,  if  great,  will 
■iKcasioD  a  considerable  lateral  thrust. 

A  vertical  shaft,  which  gives  or  receives  motion  by 
I  Deans  of  a  pulley,  has  thereby  a  lateral  pressure  brought 
iponiL 

In  Fig.  11  and  1'2  the  stress  is  compounded  of  lateral 
weaure,  arising  from  the  weight  of  the  wheels  a  and  a, 
Ind  that  of  the  shaft  itself,  and  of  the  torsion  or  twist  pro- 
fdnced  hotween  the  wheels  a  and  b. 

The  following  remarks  of  John  Hoberton,  engineer, 
I  relate  to  the  subject  of  this  Essay,  and  contain  much 
I  useful  matter  which  will  be  acceptable  to  the  reader ;  to 
[whom  it  must  be  demonstrable,  "  that  by  a  judicious  ar- 
I  Tangement  of  wheels  and  pinions,  in  many  cases  much  of 
I  lie  stress  and  friction  may  be  avoided.  This  is  a  doctrine 
Impractical  mechanics  of  very  great  importance,  when  we 
Icooeider,  that  in  many  cases  almost  the  whole  of  the  im* 
I  felling  power  is  expended  in  overcoming  the  friction  of 
|tlie  machinery. 

195.  *'  Let  A,  Fig.  1,  Plate  IV.,  be  an  overshot  water- 
Ifheel.  Let  the  line  a  b  represent  the  line  of  direction  of 
e  centre  of  gravity  of  the  water  in  the  buckets.  On  the 
■trcmity  of  this  wheel  let  there  be  a  toothed  wheel  acting 
Bto  the  pinion  b.  It  is  obvious  that,  independently  of  the 
weight  of  the  wheel,  the  whole  weight  of  the  water  will  be 
supported  by  the  axis  c,  and  the  teeth  of  the  wheel  at  d ; 
and  the  weight  which  each  will  sustain,  will  be  in  the  ratio 
of  ce  to  ed.  That  is,  by  the  principles  of  the  lever,  the 
weight  on  the  gudgeon  will  be  as  the  distance  e  d,  while 
that  on  the  teeth  will  be  as  ce\  or  if  the  radius  of  the 


184  ON   THE   SHAFTS   OF   MILLS.  [ CHAP.  II. 

toothed  wheel  were  ce^  the  teeth  would  sustain  the  whole 
weight  of  the  water,  leaviog  no  weight  on  the  gudgeon  Init 
that  of  the  wheeL  Again,  let  the  wheel  b  be  removed  to  c, 
it  is  evident  that  the  gudgeon  c,  would  have  to  sustain  the 
weight  of  the  water,  and  a  great  deal  more.  That  is, 
the  weight  on  the  gudgeon  would  be  increased  as  e({to 
ec.    The  true  relation  of  the  stress  in  the  two  cases,  is 

edi  — ^    ^^  .     So  that  it  is  evident,  the  stress  and  friction 
dc-^-ed 

of  the  gudgeon  miist  depend,  in  a  great  measure,  on  the  on 

of  the  toothed  wheel  attached  to  the  water- wheels  and  to  the 

situation  of  the  pinion  b. 

196.  '^  Again,  let  there  be  a  wheel  at  a.  Fig.  %  fixed  od 
the  end  or  middle  of  a  shaft  working  into  the  wheel  or 
pinion  b,  of  any  size. — ^Let  the  teeth  move  in  the  direcdon 
ah.     It  is  evident,  that  the  gudgeon  or  shaft  will  tend  to 
move  in  the  contrary  direction,  that  is,  in  the  direction  ci^ 
and  with  the  very  same  force  that  the  teeth  act  upon  eadi 
other,  as  action  and  reaction  are  equal  and  in  contrary  & 
rections,  and  for  the  same  reason,  the  gudgeon  of  the  wheel 
b,  will  tend  to  move  in  the  direction  e  i,  with  the  very  same 
force,  that  is,  the  same  force  as  the  action  of  the  teeth  on 
each  other.     Indeed,  in  any  single  pair  of  wheels,  of  what- 
ever form  or  construction,  the  tendency  to  break  or  bend 
the  shaft,  or  cause  friction,  is  the  same  as  the  action  of  the 
teeth  on  each  other. 

"  Now,  if  the  above  wheels  were  made  of  a  double  size,  it 
is  evident  that  the  acting  power  on  them  would  be  only 
one  half,  and  consequently,  one  half  of  the  strain  to  break 
the  shaft  or  cause  friction  *. 

197.  "  In  the  case  of  an  intervening  wheel,  the  force  or 

*  The  object  of  this  remark  is,  apparently,  to  show  the  superiority  of 
large  wheels ;  but  it  is  clear  that  the  acting  power  would  be  the  same  with 
the  same  first  mover ;  and  if  the  resistance  be  diminished  on  one  shaft,  an- 
other must  be  added  to  give  the  proposed  velocity  to  the  working  point. 


CHAP.  11.3  ON   THE   SHAFTS   OF   MILLS.  185 

tendency  to  break  the  shaft  depends  on  the  situation  of  such 
intervening  wheeL  Thus,  if  it  be  placed  in  a  direct  line 
betwixt  the  centres  of  the  conducted  and  conducting  wheels, 
as  at  A,  Fig.  3,  the  shaft  or  gudgeon  will  tend  to  move  in 
the  line  ah  or  ha^  (according  to  the  direction  of  the  con- 
ductor,) with  double  the  force  of  the  action  of  the  teeth. 

'« If  the  axis  of  the  intervening  wheel  form  a  right  angle 
with  the  axis  of  the  other  two  wheels,  the  force  to  break  the 
shaft  will  be  to  that  of  a  pair  of  single  wheels,  or  the  ac- 
tion of  the  teeth  on  them,  as  the  diagonal  of  a  square  is  to 
one  of  its  sides.  That  is,  the  direction  of  the  force,  and  its 
intensity,  will  be  represented  by  the  diagonal,  it  being  evi- 
dent, from  the  well-known  laws  of  mechanics,  that  by  the 
action  of  the  wheel  b,  Fig.  4,  No.  1,  the  centre  of  the  in- 
tervening wheel  A,  would  tend  to  move  in  the  line  a  c,  and 
by  its  action  in  the  wheel  e,  it  would  tend  to  move  in  the  line 
a  d.  Let  a  d  and  a  c  represent  the  forces  in  these  directions, 
complete  the  square  or  parallelogram,  the  diagonal  of  which 
will  both  represent  the  force  and  its  direction.  See  also 
Kg.  4,  No.  2,  and  No.  3. 

198.  "  On  the  other  hand,  if  a  wheel  be  placed  betwixt 
two  others,  as  a.  Fig.  3,  where  a  is  the  conductor,  the  teeth 
of  which  act  with  equal  force  on  each  of  the  wheels  b  and  c, 
it  is  evident  that  the  strain  is  wholly  taken  off  the  shaft,  the 
forces  being  equal  and  opposite  to  each  other  * ;  and  in  Fig.  5, 
where  a  is  supposed  to  be  the  conductor,  and  b  and  c  the 
conducted  wheels  on  which  the  teeth  on  each  bear  equally, 
— it  is  plain  from  what  has  been  already  stated,  that  the 
direction  of  the  forces  will  h^da  and  a  c,  and  letting  a  d  and 
a  c  represent  the  direction  and  intensity  of  these  forces,  and 

*  The  abaft  ought  not,  however,  in  any  case,  to  be  entirely  freed  from 
piMHin  ID  this  manner ;  because  its  motion  will  not  be  so  steady  and  re- 
giikr  M  when  there  is  some  considerable  pressure  on  the  gudgeons.  And 
dura  is  nothing  more  iigurious  in  machinery  than  a  hobbling,  unsteady 
tliii  pomt  leqmras  the  engineer  s  most  careful  attention. 


186  ON   THE   SHAFTS   OF   MILLS.  []CHAF.  It. 

completing  the  parallelogram,  we  have  the  diagonal  n  ft  to 
represent  the  compound  direction  and  intensity  of  the  force. 

199-  "  There  is  another  point  that  is  worthy  of  attention : 
that  isy  the  place  on  the  shaft  where  the  wheels  are  fixed. 
If  a  wheel  is  put  on  at  the  end  of  a  shaft  to  drive  any  other 
or  others,  it  is  clear,  that  the  whole  or  nearly  the  whole  of 
the  stress  will  be  at  the  end  of  the  shaft  or  joumaL  If  the 
wheel  be  placed  in  the  middle  of  the  shaft,  the  strain  to 
break  it  will  be  greatest  at  that  part,  but  the  force  will  be 
resisted  equally  by  each  journal.  Indeed,  on  whatever  part 
of  the  shaft  a  wheel  is  placed,  at  that  very  place  is  the 
greatest  (cross)  strain  on  the  shaft,  and  the  force  on  each 
journal  will  be  in  the  inverse  ratio  of  the  distance  of  the 
wheel  from  the  ends  of  the  shaft. 

"  In  Fig.  6,  let  a  represent  a  shaft,  either  upright  or  lying. 
If  a  single  wheel,  as  b,  fixed  upon  it,  drive  two  pinions,  as 
c  D,  directly  opposite  to  each  other,  the  shaft  or  journal  will 
not  be  afiected  thereby  *,  but  if  two  wheels  are  placed  go 
the  shaft,  driving  each  a  pinion,  as  in  Fig.  7^  both  journals 
will  be  afiected,  and  the  greatest  strain  to  break  the  shaft 
will  be  at  the  arms  of  the  wheel,  or  close  to  them,  and  be- 
twixt the  arms  and  journal.  In  this  case,  the  middle  point 
of  the  shaft,  as  at  a,  being  in  a  state  of  contrary  pressure, 
and  therefore  no  strain  on  that  part,  it  may  be  considered 
as  a  lever  on  each  side  of  a  ;  this  being  the  fixed  point, 
and  the  greatest  strain  on  the  shaft  being  at  the  wheels. 
The  strain  on  the  journal  will  be  inversely  as  the  distance 
of  the  wheels  from  the  middle  point  a,  it  being  underBtood 
that  the  wheels,  pinions,  and  resistances  are  all  the  same. 

200.  ^^  Again,  if  two  wheels  are  placed  on  a  shaft,  and 
two  pinions,  both  on  the  same  side  of  the  shaft,  the  journals 
must  support  the  strain  or  force  of  the  action  of  the  teeth 
of  both  wheels,  or  indeed  whatever  number  of  wheels  is  on 
a  shaft ;  and  working  into  pinions  on  one  side,  the  jounal 

*  See  the  note  to  Art.  I9S. 


■■] 


ON    THE    SHAFTS   OF    MILLS. 


187 


Aat  shaft  must  support  a  pressure  equal  to  the  whole  of 
r  action  into  the  teeth  of  the  wheels  or  pinions  into 
ch  they  are  connected ;  and  the  strain  on  the  shaft  to 
ik  it,  depends  on  the  situation  of  the  wheels,  as  they 

be  placed  on  the  shaft:  it  being  imderstood,  as  for- 
ly,  that  directly  at  the  place  where  the  wheels  or  pinion 
[ed  on  the  shaft,  that  it  must  support  a  pressure  equal 
le  pressure  of  the  teeth  of  the  wheel  in  its  eorrespond- 
wheel  or  pinion.  And  according  as  these  pressures 
bine,  or  act  in  a  contrary  direction  to  each  other,  so 
;  the  strain  on  the  journals,  or  tendency  to  break  or 
:  the  shaft  be. 

Hence  the  importance  of  placing  wheels  and  pinions, 
at  their  action  on  each  other  may  be  in  contrary  direc- 
{,  or  so  as  to  avoid,  as  much  as  possible,  the  strain  on 
shaft  or  journals. 

>l.  "  In  lying  shafts,  where  circumstances  may  require, 

ly  be  advisable  to  have  the  main  or  heaviest  shaft  on 
lift  of  the  wheels,  as  this  will  take  off  a  considerable 

of  their  weight  or  friction  on  the  journals.  The  other 
I  will  naturally  be  on  the  fall  of  the  wheel,  (which  is 
mod  in  this  case  much  lighter  than  on  the  main  shaft,) 
will  be  prevented  from  jolting  upwards,  from  both  its 

weight,  and  a  pressure  equal  to  that  on  the  teeth  of 

wheels,  being  supported  by  the  journals.  It  being 
lent  that  whatever  the  wheels  are,  that  the  same  pres- 
6  that  is  on  the  teeth  of  the  wheels  will  be  equallv  the 

on  both  shafts,  the  one  tending  to  increase  the  weight 
Bie  shaft  on  the  journals,  and  the  other  to  diminish  it." 
102.  Roberton,  in  the  course  of  his  business,  made  seve- 
obscrvations  on  the  foregoing  subjects.    Late  in  his  ca- 

a  case  of  this  nature  occurred,  of  considerable  import- 
at  the  flour-mill  erected  near  the  Slitt  Mills  of  Pa- 

'.     "  The  water-wheel,  16  feet  diameter,  makes  about 

*  On  llie  Itfiiiks  iif  the  Civdc. 


188  ON   THE   SHAFTS   OF   MILLS.  [CHAP.  II. 

10  or  11  revolutions  per  minute,  driving  in  general  two 
pair  of  stones ;  the  pit  wheel  about  6^  feet  diameter.  The 
consequence  was,  that  the  machinery,  firaming,  &c.,  were 
not  of  sufficient  strength  to  bear  the  force  applied  by  the 
pit  wheels,  (though  they  were  very  strong,  and  well  exe- 
cuted for  ordinary  cases,)  the  shafit,  framing,  &c'.,  were  in  a 
high  state  of  tremor,  the  machinery  working  in  a  rough 
and  straining  manner,  and  the  wooden  teeth  (5  inches 
broad,  pitch  about  4^)  could  not  stand  for  any  length  of 
time.  The  mill  was  altered,  by  enlarging  the  pinion,  to 
let  the  wheel  run  at  14  or  15  turns  per  minute ;  and  the 
pit  wheel  enlarged  about  one  foot  diameter ;  afterwards  the 
mill  wrought  very  well. 

'^  In  conunon  corn-mills,  and  many  others,  there  is  a  great 
deal  of  the  impelling  power  lost  by  the  smallness  of  the 
pinions,  &c.,  which  causes  a  great  friction  on  the  shaft  or 
spindle,  as  well  as  by  the  friction  of  the  teeth.  Were  both 
wheel  and  pinion  increased  in  diameter,  much  advantage 
would  arise,  not  only  in  saving  of  power,  but  in  the  tear 
and  wear  of  machinery.  It  is  on  this  account  that  a  com- 
mill,  constructed  in  the  double  way,  other  circumstances 
being  the  same,  performs  much  more  work  than  in  the 
single  way. 

203.  "  In  short,  more  things  of  this  nature  take  place  in 
machinery,  than  the  most  of  operative  mechanics,  or  even 
philosophers,  are  aware  of ;  and  there  can  be  no  doubt,  that, 
with  a  knowledge  of  the  affecting  causes,  a  vast  deal  might 
be  done  in  saving  power.     For  example,  in  a  cotton*mill,  the 
main  shaft  generally  makes  from  40  to  50  revolutions  per 
minute.     Let  the  weight  of  the  shafts  and  machinery,  the 
size  of  the  journals,  and  the  effect  of  the  wheels  on  these 
shafts,  and  friction  of  journals,  be  taken  into  the  account, 
in  the  one  case,  and  let  the  main  shaft  be  supposed  to  be 
reduced  to  half  of  the  former  velocity,  that  is,  from  20  to 
25  turns  per  minute,  and  let  the  strength  of  the  shaft. 


CHAP.  II.]  ON   THE    SHAFTS   OF   MILLS.  189 

together  with  a  due  proportion  of  wheels,  he  augmented, 
so  as  to  preserve  the  same  firmness  in  every  part  of  the 
machinery  as  in  the  former,  and  that  the  ultimate  part  of 
the  machinery  may  he  hrought  to  the  same  speed  as 
formerly,  it  will  he  found  on  investigation,  that  in  the  last 
case  there  will  be  a  considerable  saving  in  the  first,  or  im- 
pelling  power. 

"  It  must  not  be  lost  sight  of,  that,  by  reducing  the 
velocity  of  the  main  shaft,  a  judicious  increase  of  the 
diameters  of  the  wheels  thereon  is  absolutely  necessary. 
Indeed,  without  strict  attention  to  matters  of  that  kind,  the 
effect  of  any  alteration  that  may  be  proposed  or  made  is 
very  precarious.  A  due  regard  to  the  proper  diameter  of 
wheels,  according  to  the  work  they  have  to  perform,  is  a 
matter  of  very  great  importance ;  and  it  will  be  found,  on 
general  inquiry,  that  in  most  cases  of  machinery,  it  would 
be  prudent  to  have  the  wheels  and  pinions  of  large  dia- 
meter ;  and,  as  we  said  before,  by  increasing  their  size, 
the  force,  strain,  and  friction  on  the  shafts  and  journals 
are  diminished  in  the  same  ratio. 

"  Suppose,  in  a  mill,  that  a  range  of  Ijring  shafts,  of  80  or 
100  feet  long,  together  with  wheels,  &c.,  fixed  on  them, 
weighed  yOOOlbs  ;  the  journals  4  inches  diameter,  making 
46  turns  per  minute ;  the  surface  of  the  journal  would  at 
that  rate  move  at  about  50  feet  per  minute :  and  supposing 
that  the  friction  was  equal  to  one  third  of  the  weight,  we 
should  have  7000  ^  3  =  2333  x  50  =  11 6650 -r  44000  = 
2'66 ;  that  is,  nearly  2f  horses'  power  expended  in  over- 
coming the  friction  of  these  shafts*. 

"  Again,  were  these  shafts  and  journals  extended  to  5 

*  Thai  is,  Taluing  the  horses'  power  at  44000  lbs.  1  foot  per  minute. 
Sm  page  88,  ^ Horses  Power"  Art  108. 

The  quantity  of  frietion  is  much  overrated,  but  the  loss  of  power  would 
be  very  considerable  on  the  lowest  estimate ;  and,  therefore,  the  proper  situ- 
ation for  the  fint  mover  in  a  system  of  machmery  is  of  some  importance. 

O 


190  ON    THE   SHAFTS   OF   MILLS.  [ CHAP.  11, 

inches  diameter,  and  their  numher  of  turns  reduced  to  one 
half  of  the  former ;  that  is,  to  23  turns  per  minute,  the 
surface  of  the  journal  would  move  at  the  rate  of  31  feet 
per  minute,  and  the  weight  of  the  wheel  and  shafts  in- 
creased one  hal^  or  say,  to  10000  lbs.,  we  should  have 
10000-=-3=333Sx31  =  103323-?-44000=2-348,  that  i8» 
nearly  Q^  horses'  power ;  so  that  in  this  last  case  there 
would  be  a  saving  of  something  more  than  three  tenths  of 
a  horse's  power,  and  the  machinery  would  be,  in  every  re- 
spect, improved. 

204.  "  In  a  horse-gin,  where  a  pinion  is  driven  by  a 
toothed  wheel  on  the  gin,  the  friction,  or  strain  on  the 
journals,  depends  on  the  situation  of  the  horse-beam;  at 
any  moment  of  time  when  the  lever  or  hors^-beam  is  above 
or  below  the  pinion,  the  friction  on  the  shaft  will  be  the 
least,  and  when  in  the  opposite  direction,  the  friction  and 
force  on  the  shaft  will  be  the  greatest,  and  the  general 
friction  will  be  as  the  size  of  the  main  wheel ;  that  is,  the 
smaller  the  main  wheel  is,  the  friction  and  force  on  the 
shaft  will  be  the  greater,  and  the  larger  this  wheel  is,  the 
friction,  &c.,  will  be  the  less ;  it  being  understood  that  the 
friction  or  strain  on  the  gudgeon,  or  axle,  on  account  of 
the  reaction  of  the  teeth  of  the  wheel  is  meant^  and  not 
the  strain  on  the  shaft,  to  twist  it,  nor  any  other  friction  or 
strain  whatever. 

205.  "  In  a  water-wheel  turning  machinery,  in  many 
cases,  it  is  most  advisable  to  have  the  toothed  wheel  of  the 
same  diameter.  In  this  case,  whatever  power  or  force  is 
applied  to  the  wheel,  the  same  must  be  resisted  by  the 
teeth  of  the  wheel ;  and  it  also  follows,  of  course,  thai 
whatever  is  the  size  of  the  wheel,  or  pinion  which  is  driven 
by  the  main  wheel,  that  the  very  same  strain  is  on  the 
shaft ;  that  is,  the  same  as  on  the  teeth ;  and  the  shaft 
must  be  sufficiently  strong  to  withstand  the  pressure.  But 
another  circumstance  occurs,  that  by  increasing  the  size  of 


CHAP.  II.]  ON   THE   SHAFTS   OF    MILLS.  191 

the  pinion,  there  must  necessarily  be  increase  of  the  dia- 
meter of  the  shaft  to  withstand  the  twist.  The  shaft  by  no 
means,  however,  keeps  pace  with  the  augmentation  of  the 
wheels,  but  is  only  as  the  cube-root ;  for  instance,  a  water- 
wheel  of  16  feet  diameter  would  work  very  ill  into  a  pinion 
of  12  inches,  supposing  its  shaft  or  journal  to  be  3  inches 
diameter.  Again,  let  the  pinion  be  increased  to  2  feet 
diameter,  a  shaft  of  3f  inches  will  be  sufficient  to  with- 
stand the  twist,  the  friction  on  the  journals  will  be  much 
less,  and  the  strength  to  resist  the  strain  will  also  be  much 
increased.  General  rules  cannot  be  laid  down  with  accuracy 
in  these  things.  It  is  the  particular  circumstances  of  the 
case  that  will  guide  a  machinist  in  the  construction  of  any 
piece  of  machinery ;  and  if  he  be  not  well  acquainted  with 
these,  it  cannot  be  expected  that  his  schemes  will  be  well 
arranged. 

**  The  above  observations  are  no  doubt  at  variance  with 
the  opinions  of  those  who  are  continually  holding  up  the 
simplification  of  machinery.  For  instance,  Fenwick's 
*  Essay  on  the  Simplification  of  Machinery.'  *  In  many 
cases  it  is  prudent  to  make  machinery  of  a  more  complex 
nature  than  it  is  sometimes  constructed ;  and  in  order  that 
it  may  be  easier  driven,  that  the  tear  and  wear  may  be 
lessened,  as  well  as  the  ultimate  expense  and  trouble  of 
attending  it.''  t 

*  The  miLirimg  of  Fen\«ick  consider  friction  only,  and  arc  so  far  correct ; 
but  of  the  wear  and  tear,  and  friction  of  teeth,  he  has  taken  no  account. 
His  6th  maxim,  that  ^^  Small  wheels  are  equally  as  generative  as  large 
wheeliB,  if  the  same  ratio  of  size  be  preserved,"  is  true  only  within  certain 
limite,  (see  Art  33,)  but  these  limits  being  assigned,  his  other  maxims  hold 
tin  the  stress  on  the  moving  parts  is  less  from  a  greater  number  of  small 
wheels,  than  from  fewer  large  ones. 

t  John  Robbbton,  to  whom  the  world  is  largely  indebted,  not  merely 
for  the  fer^cMiig  dever  and  scientific  remarks,  but  for  much  of  modem  im- 
praTfOWDt,  WM,  in  his  day,  (the  beginning  of  the  present  centory,)  one  of 
thesMNl  diitingmahed  millwri^ts  or  engineers  in  Olasgow. 

o2 


192  ON   THE   SHAFTS   OF   MILLS.  [cHAP.  II. 

206.  In  the  essay  to  which  Roberton  in  the  forcing 
observations  alludes,  Mr.  Fenwick  infers,  (p.  64s)  that 
'*  the  most  perfect  machine  is  that  which  operates  with  the 
fewest  moving  parts.^*  But  Fenwick  seems  to  have  beai 
misled  here  by  a  desire  to  generalize.  Simplicity  is,  no 
doubt,  a  most  desirable  quality  in  a  machine,  provided  it 
can  be  obtained  without  making  a  sacrifice  of  power  or  of 
durability.  A  sledge  has  fewer  moving  parts,  and  in  that 
sense  is  more  simple  than  a  steam  carriage,  yet  no  one 
with  truth  could  say  that  a  sledge  is  more  perfect*. 

Nor,  perhaps,  does  the  simplicitif  of  a  machine  consist 
strictly  in  having^/few;  moving  parts.  If  the  parts  of  a  ma- 
chine be  few,  they  are  perhaps  more  easily  taken  in  by  the 
eye  at  one  view,  which  may  make  them  more  easily  compre- 
hended by  the  mind,  and  in  that  sense  be  more  simple.  But 
in  machinery,  the  kind  of  simplicity  at  which  we  ought  to 
aim,  has  more  regard  to  the  manner  of  action  than  to  the 
number  of  the  moving  parts.  Thus,  for  example,  when  a 
weight  is  to  be  raised,  if  one  wheel  worked  by  a  screw  he 
employed,  the  machine  consists  of  fewer  part^  than  two 
wheels  and  two  pinions,  applied  to  the  same  purpose.  But 
in  this  last  case,  the  manner  of  action  is  really  more  simple; 
for  the  action  and  resistance  are  directly  opposed  in  the 
same  line  ;  whereas,  in  the  case  of  the  screw,  the  action  is 
oblique,  and  experience  shows  that  it  has  much  more  fric- 
tion, and  much  less  durability. 

What  is  here  said  of  manner  of  action  is  applicahle  in 
comparing  machines,  consisting  each  of  trains  of  wheels 
and  pinions  ;  for  the  most  durable,  by  longest  maintaining 
the  true  figure  of  the  teeth,  will  ultimately  be  the  most 
simple  in  the  manner  of  its  action.  It  is  evident,  that 
when  the  teeth  become  much  worn,  that  the  manner  of 

*  The  Essay  by  Fenwick  here  alluded  to,  is  wholly  confined  to  the  <^ 
plicatum  of  wheel- work ;  consequently  the  remarks  in  this  and  the  follow- 
ing paragraph  are  not  applicable  to  his  Essay. 


....] 


ON    THE    SHAFTS    OF    MILLS. 


I'J^ 


becomes    proportionably   more    oblique    and    less 

lie.     Respecting  the  durability  of  the  wheel-work  of 

we  may  refer  the  reader  to  Art.  UX). 

It  can  be  of  very  little  use  to  give  rules  for  the  dia- 

of  gudgeons,  or  for  the  strength  of  shafts,  unless 

be  accompanied  with  some  method  of  estimating  the 

ng  force.      Our  author  has  not  touched  upon  this 

^  of  his  subject ;  and  therefore  we  have  inserted  Ro- 

berton's  remarks,  with  the  ^iew  that  the  information  which 

convey,  respecting  the  stress  on  shafts  and  gudgeons, 

be  studied  in  their  proper  order ;  we  shall  add  some 

ional  inquiries  to  these  articles, 

'.  Let  AB,  Plate  [IV.  A.]  Fig.  4.,  represent  a  shaft, 

one  wheel  at  d,  and  another  at  c  ;  these  wheels  being 

any  size  whatever.     If  a  power  act  upon  the  wheel  d  at 

thi!  point  F,  and  the  resistance  be  at  w,  the  stress  arising 

from  these  forces  will  cause  a  pressure  on  both  the  gudgeons ; 

the  line  wp  being  drawn,   cutting  the  axis  at  some 

E ;  the  stress  upon  the  shaft  and  gudgeons  will  be 

same  as  if  a  force  equal  to  the  power  and  resistance 

together,  were  applied  at  the  point  e.     Hence  it  is  clear. 

It  when    the   wheels   differ    considerably    in    size,    the 

IpoD  next  the  smaller  wheel  will  have  to  sustain  the 

iter  part  of  the  stress. 

08.  But  if  the  resistance  were  at  w,  the  power  and  re- 
ttice  in  this  case  being  at  the  same  side  of  the  shaft, 
pressure  will  be  downward  on  one  gudgeon  and  up- 
i  upon  the  other.  For  the  descending  power  p  is  re- 
sd  by  the  support  of  the  gudgeon  at  a,  and  by  the  re- 
at  w ;  but  the  power  not  falling  between  these 
it  will  tend  to  raise  the  gudgeon  b. 
KJ.  Again,  if  the  resistance  be  at  the  point  c  on  the 
or  under  side  of  the  wheel  c,  the  pressure  at  the 
u  will  be  wholly  in  a  lateral  direction ;  conse- 


194  ON   THE   SHAFTS  OF  MILLS.  [^CHAP.  H. 

quently,  when  the  impelling  power  moves  with  the  wfaed, 
the  stress  on  the  gudgeons  will  vary  considerahly  both  iB 
intensity  and  direction. 

If  the  plan  of  the  shaft  and  wheels  he  drawn  to  a  scale, 
it  will  be  easy  to  compute  the  pressures  in  these  difimnt 
cases,  and  to  compare  them ;  and  perhaps  a  young  machinist 
will  feel  some  pleasure  in  such  comparisons,  where  he 
would  have  been  fearful  of  engaging  with  a  mass  of  algebra. 

Case  1.  The  power  and  resistance  being  at  opposite 
sides  of  the  shaft.  Draw  the  line  a  b  in  the  middle  of  the 
shaft,  and  also  draw  the  line  wp.  Then  to  find  the  stresB 
upon  the  gudgeon  b,  we  shall  have  a  b  :  a  e  : :  power  added 

to  the  resistance  :  stressonthe  gudgeon  b;  or ^ ^^ 

AB 

the  stress  upon  the  gudgeon  b.  Also  ab  :  be  ::  power 
added  to  the  resistance  :  stress  on  the  gudgeon  a  = 
be  X  (p  +  w) 

■  ■  • 

ab 

Case  2.  The  power  and  resistance  being  at  the  same 
side  of  the  shaft.  Draw  the  lines  Ate  and  pb  which  cut 
one  another  at  f.  Then,  the  line  pb  mav  be  considered  a 
lever  with  its  fulcrum  at  f  ;  and  to  find  the  stress  neces- 
sary to  keep  the  gudgeon  b  down,  we  have  bf  :  pf  ::  power 

at  p  :  stress  at  b  = =  the  stress  on  the  gudgeon  B. 

B  F 

This  stress  will  obviously  be  opposed  to  the  weight  of  the 
shaft  and  wheels.     Also,  af  :  wf  ::  the  resistance  at  w  : 

stress  on  the  gudgeon  at  a,  = .     This  stress  will  be 

AF 

to  add  to  the  stress  from  the  weight  of  the  shaft  and  wheels. 

Here  it  may  be  remarked,  that  it  is  desirable  that  the 
greater  pressure  on  any  gudgeon  should  always,  when  con- 
venient, be  in  the  direction  of  gravity,  to  prevent  the  un- 
pleasant jolts  which  take  place  when  the  machine  is  pat 


ON    THE    SHAFTS    OF    MILLS. 

So  motion,  when  the  pressure  is  upwards  upon  any  of 
igadgeons. 

210.  But  the  preceding  cases  suppose  that  the  power  at 
and  the  weight  or  resistance  at  w,  are  parallel,  they  are, 
wever,  oft^n  ohlique  in  respect  to  one  another.  Let  k, 
p.  5,  (i,  and  7.  on  a  plan  of  such  wheels,  be  the  axis ; 
I  p  be  that  point  in  the  eircnmference  of  one  of  them 
Bpe  the  power  acts  in  the  direction  d  p  ;  and  let  w  be 
i  point  where  the  resistance  acts  in  the  direction  dw. 
ben  DE  will  be  the  direction  of  the  stress  upon  the  axis, 
d  if  D  c  be  made  proportional  to  the  resistance ;  or  u  A 
opcH'tional  to  the  power ;  and  if  the  parallelogram  Dcab 
>C4nnpleted,  «d  will  be  proportion^  to  the  whole  stress 
on  the  axis,  which  is  obviously  greatest  in  Fig.  5,  and 
It  in  Fig.  f).  Now,  make  cd  perpendicular  to  de,  tlien 
i  is  the  pressure  on  the  axis  caused  by  the  resistance  at 
)  circumference  of  the  lesser  wheel ;  and  ad  will  be  the 
on  the  axis  from  the  power  at  the  circumference 
the  large  wheel.  But  it  must  be  remarked,  that  when 
8  direction  of  the  stress  on  the  axis  falls  between  the  di- 
tion  <rf  the  power  and  that  of  the  resistance,  as  in  Figs. 
Hid  7 ;  the  whole  stress  will  he  either  in  the  direction 
1)  as  in  Kg.  5,  or  ED  as  in  Fig.  7i  and  proportional  to 
I.  Whereas,  if  the  directions  of  the  power  and  rosist- 
Ke  be  both  on  the  same  side  of  the  axis,  as  in  Fig.  0, 
I  »-ill  be  the  pressure  on  the  axis  at  the  place  of  the  small 
leel  w,  in  the  direction  e  d,  tending  to  raise  the  axis ; 
Q  rfa  the  pressure  on  the  axis  at  the  large  wheel  p,  in 
B direction  de  ;  or  opposite  to  the  pressure  at  the  wheel 
i  while  iiD   is  the  difference   between  these  pressures, 

the  whole  tendency  of  the  axis  to  rise. 
The  figures  are  all  drawn  to  the  same  size,  and  the 
T  and  resistance  being  represented  by  equal  lines  in 
figurt',  the  ad-vantage  or  disadvantage  of  any  particu- 


196  ON   THE   SHAFTS   OF   MILLS.  [CHAP.   ^ 

lar  construction  will  be  seen  by  inspection ;  and  the  rea^ 
may  easily  multiply  examples,  by  drawing  more  figures. 

The  relation  and  intensity  of  the  pressures  might  hsLr 
been  shown  in  a  more  elegant  manner  by  the  arithmetic 
sines,  but  perhaps  not  so  satisfactorily  to  the  minds  of  mo 
of  my  readers.  And  when  the  rule  and  compasses  are  : 
the  hand,  the  relations  are  more  easily,  and  with  less  ris 
of  mistake,  ascertained  by  drawing  the  wheels  and  axis  a 
shown  in  the  figures ;  as  it  merely  requires  a  little  know, 
ledge  of  the  composition  and  resolution  of  forces. 

Thus,  make  d  c,  from  a  scale  of  equal  parts,  equal  to 
the  number  of  cwts.  in  the  resistance  at  w,  and  make  ac 
parallel  to  the  power  pd,  and  cd  perpendicular  to  de. 
Then  d  d,  measured  from  the  same  scale,  is  the  stress  at 
the  centre  of  the  wheel  w  in  cwts.  and  ad  the  stress  in 
cwts.  at  the  centre  of  the  wheel  p. 

The  stress  from  the  resistance  and  power  being  deter- 
mined, that  which  is  caused  by  the  weight  of  the  shafts 
and  wheels  themselves  will  be  easily  calculated.  See  Art 
245  and  the  following  articles. 

The  case  of  lateral  stress  being  the  most  simple,  we 
shall,  in  the  first  place,  examine  it,  confining,  for  the  pre- 
sent, our  attention  to  the  gudgeons  only.  The  bodies  of 
shafts  will  be  afterwards  considered. 


CHAPTER  III. 


It  TEg  STRENGTH  ( 


>  ODDOBONS  WHBRB  THE  STBESS  IS   PRODUCED  BIT 
LATERAL  PRESSURB   DNLV. 


The  gudgeons  having  all  the  weight  on  the  shafit 
mpport,  ought  to  be  made  sufficiently  strong  for  that 
while,  to  avoid  unnecessarj'  friction,  they  should 
ifflade  as  small  in  diameter  as  possible,  conEistently  with 

lent  strength  and  durability. 
■When  we  are  able  to  determine  the  diameters  of  the 
gudgeons,  or  journals,  this  serves  as  a  foundation  for  the 
Jroportions  of  the  other  parts  of  the  shafts. 

ilthough  wTought  iron  will  bear  a  greater  weight  than 
iron,  yet  cast  iron  being  not  only  cheaper,  but  much 
»  easily  formed  into  convenient  shapes,  gudgeons 
Bow  most  commonly  made  of  that  material.  We  shall, 
refore,  in  the  first  place,  confine  our  attention  to  the 
Hbeters  of  gudgeons  made  of  cast  iron.  Here  it  may 
^per  to  state  the  following  proposition  : 
ill  Prop.  I — Solid  cifUnders  of  the  same  letigtk  have 
lateral  strength  as  (lie  cube  of  their  diameters  ',  Jbr 
ffieral,  the  lateral  strength  of  any  pieces  of  iron  or 
whose  sections  are  similar  Jigures,  are  as  the  cubes 
fe  timilar  sides  of  t/ie  sections. 
That  is,  if  a  gudgeon  of  two  inches  be  sufficient  to  sup- 

Soe  Bmemou'ti  4to  edition,  prop.  67,  cor.  2.    Gregory 'a  Mochanics, 
iuL  I7U.  cor.  3. 


198  ON   THE   SHAFTS   OF   MILLS.  [CHAP.  lU. 

port  a  certain  weight,  a  gudgeon  of  four  inches  will  sap- 
port  eight  times  as  much. 

From  this  law,  it  is  evident,  that  were  all  gudgeons 
made  of  iron  of  the  very  same  quality,  knowing  the 
strength  sufficient  in  any  one  case,  it  would  be  easy  to  cal- 
culate what  it  should  be  in  any  other  case. 

But  as  there  is  a  great  variety,  in  point  of  strength,  in 
different  kinds  of  iron ;  Welch  cast  iron,  for  instance, 
being  stronger,  as  some  think,  by  one  fourth,  than  that 
made  in  Scotland ;  it  is  prudent  to  calculate  upon  the 
weakest.  Mr.  Banks  observes  %  that,  '^Iron  is  much 
more  uniform  in  its  strength,  than  wood ;  yet  it  appears 
that  there  is  some  difference  in  different  kinds  of  ore,  cr 
iron-stone ;  there  is  also  a  difference  from  the  same  for* 
nace,  perhaps  owing  to  the  degree  of  heat  which  it  has 
when  poured  into  the  mould.'* 

213.  The  strength  of  a  gudgeon  is  limited  by  the  strain 
it  will  bear  without  permanent  derangement  of  its  struc- 
ture, for  the  length  is  always  so  small  in  regard  to  the  diar 
meter  that  the  flexure  will  be,  in  all  practical  cases,  insen* 
sible. 

The  calculated  stress  should  include  every  kind  of  force 
acting  on  the  axis  or  shaft ;  and  the  diameter  should  he 
determined,  so  that  the  gudgeon  would  be  capable  of  re- 
sisting the  whole  stress  if  it  were  thrown  upon  the  extreme 
point  of  its  bearing,  (see  Art.  217- ) 

When  w  is  the  utmost  amoxmt  of  the  stress  in  cwts.,  and 
/  the  length  of  the  gudgeon  in  inches,  from  the  shoulder  t^ 
the  extreme  point  of  bearing,  it  is  shewn  (Essay  on  Cas 

Iron,  Art.  138.)  that  Cli^ZJiJ^*  ==  rf;  the  diameter  o 

5 

the  gudgeon  in  inches.     Or,  0*42  ( w  /)  *  =  rf. 

214.  But  an  allowance  should  be  made  for  wear,  which 
will  be  nearly  directly  as  the  stress,  and  inversely  as  the 

*  Banks  s  Power  of  Machines,  p.  94. 


ON   THE    SHAFTS    OF    MILLS.  19!) 

len^h  of  the  gudgeon's  bearing ;  consequently  the  length 
of  tiie  gudgeon  should  be  greater  in  the  same  ratio  as  the 
stress  is  greater.  And  till  some  more  certain  principles 
of  jjrojKjrtioning  the  gudgeons  of  a  machine  so  as  to  be  of 
equal  duration  shall  be  found,  we  may  allow  one  fifth  of 
tbe  diameter  as  a  provision  against  wear  where  no  gritty 
substance  is  likely  to  affect  it,  and  one  third  in  all  cases 
irhere  the  gudgeons  are  exposed  to  gritty  matters. 

RcLE — In  the  former  case,  the  rule  will  become  0*5 
(«/)!  =d.  That  is,  multiply  the  stress  in  ewts.  by  the 
length  of  the  gudgeon  in  inches,  and  the  cube  root  of  the 
product  being  multiplied  by  0-6,  will  give  the  diameter  of 
ik  gudgeon  in  inches. 

WTicn  a  gudgeon  is  likely  to  wear  much  from  the  nature 
i  the  machine  or  its  particular  situation,  multiply  the 
•.  root  of  the  product  by  O'fi,  instead  of  0-5.  Gud- 
s  of  water-wheels  may  be  included  in  the  class  which 
«  exposed  to  considerable  wearj 

I K  the  stress  on  one  gudgeon  be  equal  to  the  weight  of 
|e  wheel,  and  the  wheel  be  at  the  middle  point,  that  part 
■  the  stress  which  is  produced  by  the  action  of  the  moving 
Heer  and  resistance,  must  be  considered  equal  to  half  the 
ight  of  the  wheel ;  for  only  half  the  weight  will  bear  on 
K  of  the  gudgeons  in  this  case.  But  it  often  happens 
it  the  wheel  is  considerably  nearer  to  one  bearing  than 
B  other,  and  in  such  cases,  the  rule  of  the  author  woidd 
B  likely  to  mislead. 

[  Since  our  author  has  given  it  as  a  general  principle,  that 
K  diameter  of  a  gudgeon  should  be  equal  to  the  cube 
f^»t  of  the  weight  supported  in  cwts.,  it  will  be  desirable 
">  compare  our  rule  witli  that  principle ;  first  assuming 
Ml  the  weight  ia  actually  equal  to  the  stress  upon  the 
IRlgeon.  Now,  in  tbat  case,  the  rules  will  be  the  same 
D  0-5/1  =  1,  or  /  =  2  1  =  1"20.    And  in  the  second  rule, 


200  ON   THE   SHAFTS   OF   MILLS.  [CHAP.  UI. 

when /=  (— )   =1*185  inches.     Therefore  whenever  the 
•6 

length  of  the  gudgeon  exceeds  ahout  1  inch  and  ^,  the  rule 
gives  the  diameter  too  smalL  Again,  if  we  take  the  actual 
stress  to  be  only  half  the  weight  of  the  wheel  in  cwts., 
which  is  clearly  in  all  cases  less  than  the  real  stress,  the 
above  numbers  should  be  multiplied  by  the  cube  root  of  % 
which  will  show  that  the  author's  rule  becomes  in  defect 
again  when  the  length  exceeds  1*587  inches. 

We  shall  proceed  in  this  inquiry,  on  principles  siinilar 
to  those  which  have  been  followed  in  the  Inquiry  respect- 
ing the  Strength  and  Durability  of  the  Teeth  of  WhedSf 
namely,  by  taking  a  number  of  cases  from  mill- work  in  ac- 
tual use,  and  drawing  inferences  from  them.    This  metbod 
of  making  inferences,  by  collecting  and  arranging  facts  re- 
specting miU-work,  being  much  safer  and  more  usefiilf 
than  founding  calculations  upon  experiments,  often  made 
on  a  small  scale,  and  under  circumstances  very  different 
from  those  which  occur  in  practice  with  machinery.    We 
begin  with  considering  the  gudgeons  of  water-wheels. 


SECTION  II. 


OP   GUDGEONS   OP   WATER-WHBELS. 


215.  In  the  following  table,  water-wheels  of  various 
weights  are  collected,  and  the  diameters  of  the  gudgeons 
in  actual  use,  stated.  The  weights  of  the  cast  iron  wheels 
were  found,  from  the  weight  of  the  castings,  &c.,  of  which 
they  are  composed.  The  wooden  wheels  are  estimated, 
making  allowance  for  the  wood  becoming  heavier  by  being 
soaked  in  water.  We  are  aware,  however,  that  besides  the 
mere  weight  of  the  wheel,  other  circumstances  should  be 


.  .„.] 


ON    THE   SHAFTS    Of    MILLS. 


201 


&en  into  account',  such  as  the  weight  of  water  in  the 
buckets,  and  the  pressure  brought  on  the  gudgeon  by  the 
resistance  of  the  work,  &c.  But  we  shall  follow  the  ge- 
neral result  of  those  cases  only,  in  which  the  gudgeons 
havo  been  found  sufficiently  strong,  and  we  apprehend 
that  those  extraneous  causes  on  the  one  hand,  will  not 
affect  our  rules  more  than  the  different  qualities  of  cast 

BwiU  the  strength  of  gudgeons  on  the  other  handt. 

fhe  water-wheels  in  the  table  were  all  in  the  middle  of 

ir  respective  shafts.     Both  gudgeons  of  each  wheel  had, 

lefore,  equal  stress. 

216.  Description  of  the  first    Table  of   Gudgeons  of 

'dtr-telieels. 

Column  1.  contains  letters  to  distinguish  the  wheels. 

2.  shews  the  material  of  which  the  wheel  is 
made. 

3.  the  diameter  in  feet, 
■i.  the  width  in  feet. 

5.  the  kind  of  wheel. 

6.  the  diameter  of  the  gudgeon  in  inches. 

*  See  Encyclopfedia  Biitannica,  ulicle  Rotation. 

*  These  remarks  of  Bobcrtaon  Buclianan  render  it  necessary  to  say  a 
■ule9  founded  on  empirical  principles,  and  particularly  when 

ire  not  minutely  detuled  for  those  cases  on  which  the 
I  m  founded. 

It  rule  far  the  strength  of  gudgeons  in  the  text,  supposes  the  stress  to 
llnjti  proportional  to  the  weight  of  the  wheel,  but  thia  supposition 
triy  ever  corresponds  «nth  the  truth  ;  consequently,  in  the  cited  proc- 
t,  if  the  stress  was  not  the  greatest  possible,  in  regard  to  the  weight 
;lb  wheel,  and  the  meta!  of  an  inferior  quahty,  Uiis  rule  may  lead  us 
Iwrious  errors.  But  the  practical  eases  ore  not  described,  and  therefore 
>t  judge,  from  any  thing  in  the  teat,  of  the  safety  of  the  rule. 
B  mlc  ia  to  be  formed  by  any  process,  every  cause  of  stress  should 
imiiilered,  and  where  siinphcity  is  desirable,  the  stress  should  be  rcpre- 
d  by  a  quantity  which  is  certain  to  equal  it,  even  in  an  extreme  case. 
(Uaot  err  greatly,  if  the  error  be  always  on  the  dde  of  strength. 


302  ON  THB  SHAFTS  OF  HILLS.  f {»AP.  m. 

Column  7.  the  weight  of  £ome  of  the  wheels,  in  tou 
and  cwtB. 

8.  the  weight  of  some  of  the  wheels,  in  cwti. 
and  qrs. 

9.  contains  the  cube  root  of  the  weight 
The  use  of  this  column  is  to  compare  the  several  gud- 
geons with  the  law  contained  in  IVop.  I.  For,  were  tH 
the  gudgeons  duly  proportioned  to  the  weight  they  hafe  to 
sustain,  they  would  be  to  one  another  as  the  cube  roots  (tf 
their  weights. 

TABLE   I. GUDGEONS   OF   WATER-WHEELS. 


1 

2 

3 

i 

5 

G 

7 

8 

9 

y 

i 

'tl 

wrighiof 

Wright  of 

Wheel 

^ 

3 

i 

whHilllll 

fheeli  tn 

node  of 

Elod. 

lontind 

cwu.u>d 

Df  wdshl 

II 

1 

cint. 

*""■ 

Incou. 

A 

Wood 

24 

12 

Ovcrshol 

7 

23 

14 

474 

7-796974 

B 

Cart  iron 

16 

6 

ditto 

7 

12 

Q40 

3 

6-214464 

C- 

Caitiron 

16 

8 

ditto 

6i 

16 

io 

930 

6-91IM23Th?MJ(~ 

D 

Caalimn 

1 

wheel  *nd 

[^ 

4i 

ditto 

24 

480 

7 -829736  ouu*. 

buckets 

f 

1 

E 

Wood 

16 

fl 

ditto 

6 

10 

211 

5-95334 1| 

F 

Wood 

121 

7 

ditto 

6 

5 

14 

114 

4-8488081 

G 

Wood 

3-2 

11 

ditto 

10 

H 

Hi 

10 

ditto 

7 

Wood 

« 

ditto 

8 

K 

Rr 

i» 

lO 

ditm 

10 

•  Tbe  holhn  ihuft  oT  C 


OBIBRVAIIONS  I 


I   THB   PtBST  TABLB  OF  QUDOEOHS. 


217.  Particular  care  should  be  taken  that  the  axis  of 
the  gudgeon  be  exactly  in  a  line  with  the  axis  of  the  shaft 
which  it  supports,  otherwise  the  motion  will  be  imequal, 
and  at  one  part  of  the  revolution  the  stress  will  be  thrown 


CHAP.  III.]  ON   THE   SHAFTS   OF   MILLS.  203 

to  the  point  of  the  gudgeon;  this  would  endanger  its 
breaking,  more  particularly  if  very  long,  though  otherwise 
sufficiently  strong*.  In  the  case  h,  we  have  an  instance 
of  a  gudgeon  breaking,  from  being  made  too  long.  In 
practice  it  is  a  good  method  to  turn  the  gudgeons  of  a 
wooden  shaft,  after  they  arc  fixed  in  their  places,  a  second 
time,  in  order  to  render  them  quite  true. 

218.  From  comparing  the  diameters  of  the  gudgeons 
(column  6)  with  the  cube  root  of  the  weight  in  cwts.  of  the 
wheel,  (col.  9,)  it  will  be  found  that  they  approach  one 
another.  In  other  words,  the  •  cube  root  of  the  weight  in 
cwts.  is  nearly  equal  to  the  diameter  in  inches.  In  the 
case  c,  the  gudgeon  broke  only  from  being  a  bad  casting, 
although  it  is  smaller  in  proportion  to  its  weight,  than 
most  of  the  other  cases.  Since  it  was  renewed  of  the  same 
size,  it  has  continued  to  support  its  work.  When,  there- 
fore, we  can  ascertain  the  weight  of  a  water-wheel,  we 
have  a  very  simple  rule  for  finding  the  diameter  which  the 
gudgeon  ought  to  have. 


RULE  I. 

219.  The  cube  root  of  the  weight  of  a  water-wheel^  in 
hundredweights^  is  nearly  equal  to  the  diameter  in  inches 
of  a  cast  iron  gudgeon  sufficiently  strong  to  support  such 
wheelf. 

We  say  nearly^  it  being  evidently  most  prudent  to  make 
the  gudgeon  a  little  more  rather  than  less  in  diameter,  and 
to  make  aUowance  for  wearing. 

*  The  possibility  of  such  a  cause  of  failure  should  be  guarded  against  in 
proportioning  the  strength  of  a  gudgeon.    See  Art.  213. 

t  In  water-wheels,  the  stress  is  not  proportional  to  the  weight  of  the 
wlieeL  See  Art  212 — ^215  inclusive,  where  more  correct  principles  are 
iiifiettigited.    Alao  tee  the  cantions  in  note  to  Art.  215. 


S04    .  ON   THE   SHAFTS   OF   MILLS.  [CHAP.  10. 


EXAMPLE. 

Suppose  a  water-wheel  to  weigh  12  tons,  0  cwt,  3  qn. 
what  ought  to  he  the  diameter  of  a  cast-iron  gudgeon,  suf- 
ficiently strong  to  support  the  wheel  ? 

12  tons  =  240  cwt.  3  qrs. 

The  cuhe  root  of  24075  =  6*221  Answ.  That  is,  the 
diameter  of  the  gudgeon  should  not  he  less  than  6*^  dia- 
meter. It  ought  to  be  rather  more,  to  allow  for  wearing, 
&c.     See  B,  in  the  first  table  of  gudgeons. 

220.  As  the  weights  of  wooden  water-wheels  cannot  be 
accurately  known,  without  a  good  deal  of  calculation,  it  is 
desirable  to  have  some  more  ready  method  for  practical 
purposes. 

The  weights  of  overshot,  or  bucket  water-wheels,  will 
be  to  one  another  nearly  as  their  circumferences,  or  dia- 
meters and  breadth. — ^We  say  nearly^  because  the  arms  will 
make  large  wheels  heavy  in  rather  a  greater  proportion. 
Hence  the  following  rule  is  formed  on  the  direct  proportion 
of  the  sole  and  buckets,  adding  one-half  of  the  diameter 
increased  in  the  duplicate  ratio  or  square  of  the  diameter. 


RULE  IL 


For  fcooden  water-wheels^  multiply  the  diameter  in  fid 
by  the  width  also  in  feet,  to  which  add  the  square  of  half 
of  the  diameter.  The  cube  root  of  the  sum  will  be  nearly 
equal  to  the  diameter  of  the  gudgeon  in  inches. 


EXAMPLE. 


Suppose  a  wiioden  water-wheel  1 2  feet  diameter  and  7 
feet  wide,  (^see  e  in  Table  !!•  of  Gudgeons.) 


CHAP.  III.J  ON   THE    SHAFTS    OF    MILLS.  ^).5 

12  X  7  =  84 
The  square  of  6  «  36 

ISO  the  cube  root  of  which  =  4'9S24.24; 
that  is  the  gudgeon  should  not  be  less  than  about  5  inches 
diameter. 

IZPLAHATIOH   OP  TABLE   II.   OF  WATBB-WHSBL8. 

2S1.  All  the  columns,  except  No.  10,  are  the  same 
as  in  Table  X,*  The  colunm  10  shews  the  result  by 
Rule  II. 

TABLE   II GUDGEONS   OF    WATER-WHEELB. 


i 

t 

DlDin 

Wherf 

s 

i 

KIpd. 

w^hi  or 

1 

lUnuirlu. 

A      Wood 

24 

~ii 

Overahol 

7 

23 

14 

474 

7-796974 

7-559525 

B  Out  iron 

16 

16 

ditto 

7 

12 

2 

24^ 

6-231678 

C  C«t  iron 

16 

8 

ditto 

»l 

16 

10 

330 

6-910423 

DOW  iron 

32 

4i 

ditto 

9 

SO 

400 

7-368063 

nfaeeluid 

buclieU 

K      Wood 

12 

7 

ditto 

C 

5 

14 

114 

4-848808 

4-932424 

F     Wood 

S& 

11 

diao 

10 

9-471647 

G     Wood 

21 

10 

ditto 

7 

6-839903 

H     Wood 

28 

6 

ditto 

8 

7-140037 

I      Wood 

16 

9 

ditto 

6 

io 

211 

s-a^i 

5-924991 

K     Wood 

18 

10 

ditto 

10 

.. 

6-390676 

r    IBON  ODDOEONS  FOB  VABIom  FUBFOaSS. 


2S2.  Taking  it  for  granted  that  the  cube  root  of  the 
weight  of  a  water-wheel,  in  hundredweights,  is  nearly 
equal  to  the  diameter  in  inches  of  a  cast  iron  gudgeon, 
sgfficiently  strong  to  support  such  wheel,   the  following 

*  In  then  table*  Uh  wd^t  of  the  wheel  b  U  diffeicot.  We  mippoae  the 
Int  taUe  to  1w  the  oomct  <ma,  bat  the  difference  ia  iacondderable. 


206 


ON   THE   SHAFTS   OF   MILLS*  [CHAP.  in. 


table  of  the  diameters  of  gudgeons,  and  the  weights  which 
they  may  be  supposed  to  sustain,  is  formed.  It  may  be  of 
use  for  finding  the  diameters  of  journals  in  all  cases  of 
stress  arising  from  lateral  pressure,  such  as  grindstone, 
intermediate  spindles,  &c.  where  the  pressure  can  be  ascer- 
tained, as  well  as  water-wheels*. 

EXPLANATION  OF   THB   TABLE  OF  CAST  *IBON   6UDOB0N& 

223.  Column  1  contains  the  diameter  in  inches,  firom  1 
to  11  inches. 

Column  2  contains  the  cube  of  that  diameter,  or  the 
hundredweights  which  the  gudgeon  may  sustain. 

N.B.  We  have  already  remarked,  (Art.  219f)thatit 
would  be  most  prudent  to  make  the  gudgeon  a  little  more 
in  diameter  than  the  cube  root  of  the  hundredweights. 


TABLE   OF   CAST   IRON    GUDGEONS. 


TX»                A             •— 

Cube  of  dBameter,  or 

Diameter  in 
inches. 

Cube  of  diameter,  or 

Diameter  m 

cwts.  wnioh  toe  gua* 

cwtB.  which  the  f^nd- 

m  WKXi^mM^Cm 

geon  may  austun. 

m^m,\^mm^^9^ 

geon  may  sustam. 

1- 

1- 

6-25 

244-140625 

1-25 

1-953125 

e-b 

274-625 

1-5 

3-375 

6-75 

307-546875 

1-75 

5-359375 

7- 

343- 

2- 

8- 

7-25 

381-078125 

2-25 

11-400625 

7-5 

421-875 

2-5 

15-625 

7-75 

465-483375 

2-75 

20-796875 

8- 

512- 

3- 

27- 

8-25 

561-515625 

3-25 

34-328125 

8-5 

614125 

3-5 

42-875 

8-75 

669-921875 

3-75 

52-734375 

9- 

729- 

4- 

64- 

9-25 

781-453125 

4-25 

76-765625 

9-5 

875-375 

4*5  • 

91-125 

9-75 

926-859375 

^•75 

107-171875 

10- 

1000- 

5- 

125- 

10-25 

1076-890625 

5-25 

144-703125 

10-5 

1157-625 

5*5 

166-375 

10-75 

1242-296875 

5-75 

190-109375 

11- 

1452- 

6- 

216- 

*  See  Art.  212. 


CHAP*  IIL]  on   the    shafts    OF   MILLS.  207 


USE  OP  THE  TABLE. 


EXAMPLE  L 


324.  Suppose  a  cast  iron  gudgeon  of  5f  inches  diameter, 
what  weight  of  a  water-wheel  would  it  he  capable  of  sus- 
taining? 

In  the  first  column  find  5*^5.  Opposite  to  which  will 
be  found  190*  109375  hundredweights,  which  is  rather 
more  than  9i  tons,  which  is  the  answer.  But  in  practice/ 
it  would  perhaps  be  prudent  not  to  load  this  gudgeon  with 
more  than  nine  tons. 


EXAMPLE  IL 

Suppose  a  grindstone  weighing  \5\  hundredweights, 
required  the  size  of  a  gudgeon  sufficient  to  sustain  this 
weight  Look  in  the  second  column  for  the  nearest  weight 
to  that  given,  which  will  be  found  to  be  15*625,  opposite 
to  which,  in  the  first  column,  is  9,\  inches,  the  diameter  of 
the  gudgeon  required. 

But  in  practice  the  spindles  of  grindstones  are  com- 
manly  of  wrought  iron,  which  metal  we  shall  presently 
consider  as  applied  to  gudgeons. 


SECTION  IV. 

or  MALLBABLE  OB  WBOUOHT   IBON  OUDOBOirS. 

225.  Professor  Robison  states*,  that  the  cohesive  force 
of  a  square  inch  of  cast  iron  is  from  40,000  to  60,000  lbs., 
wionglit  iron,  from  60,000  to  90,000  lbs. 

In  the  year  1795,  Buchanan  had  occasion  to  substitute 
cast  iron  gudgeons  for  those  of  wrought  iron,  and  made 

*  KnejdopndiB  Britannica,  article  Strength  of  Materials,  40. 

p2 


208  ON    THE   SHAFTS   OF   MILLS.  []CHAP.  HL 

some  experiments  on  those  metals,  firom  which  he  drew  the 
following  inference :  that  gudgeons  qf  the  same  size^  of 
cast  and  of  wrought  iron^  in  practice^  are  capable^  at  a 
medium^  qf  sustaining  weights  without  flexure^  in  the  pro- 
portion qf9  to  14*. 

Taking  it  for  granted  that  this  proportion  is  near  the 
truth,  we  may  find  the  diameter  which  any  wrought  iron 
gudgeon  ought  to  have  when  its  lateral  pressure  is  given, 
in  the  following  manner : 

226.  1.  Find  the  diameter  which  a  cast  iron  gudgeon 
should  have  to  sustain  the  given  pressure,  then  say,  as  14 
is  to  the  cube  of  the  diameter  of  the  cast  iron  gudgeon,  so 
is  9  to  the  cube  of  the  diameter  of  the  wrought  iron 
gudgeon. 

2.  The  root  of  this  last  number  gives  the  diameter  re- 
quired of  the  wrought  iron  gudgeon. 

*  According  to  Tredgold's  experiments,  the  stiffiiess  of  good  cut  iron  b 
to  that  of  good  English  malleable  iron  as  1  is  to  1*3  nearly.  (Eanjon 
Cast  Iron,  Art.  Iron.)  But  the  stifihess  of  malleable  iron  is  much  increued 
by  hammering,  &c.;  and  in  his  trials  some  pains  were  taken  to  obtain  the 
resistance  unhammered,  which  most  probably  causes  the  difference. 

It  is  further  necessary  to  observe,  that  in  calculating  the  Btrength  of 
gudgeons,  the  resistance  to  permanent  alteration  is  the  proper  mcasoiei  be- 
cause they  are  too  short  to  admit  of  sensible  flexure ;  and  the  strain  whidi 
produces  permanent  alteration  in  malleable  iron  is  only  1*12  times  that  pro- 
ducing a  like  alteration  in  cast  iron. 

Hence  the  diameter  of  a  malleable  iron  gudgeon  should  be  0*963  times 
that  of  a  cast  iron  one  to  bear  the  same  stress.  For  if  a  be  the  diametCT  of 
the  cast  iron  gudgeon,  and  b  the  diameter  of  the  \iTOught  iron  one ;  then 
a^  X  1    =  ^^   X    1*12   when  their  strengths  are  equal;   consequently 

n2  =*'' °' (TT2)*  =  *' ^"*  (T^*  =  »•««»'»  =  *• 

The  proportion  given  by  our  author  is  about  0*863  azz.by  bat  his  propor- 
tion is  made  from  the  relative  stiflfhess,  while  the  rule  applies  to  the 
strength,  and  therefore  it  is  not  correct 


CBAP.in.]  ON   THE   SHAFTS   OF   MILLS.  209 

EXAMPLE. 

Suppose  the  lateral  pressure  to  be  125  hundredweights, 
the  cube  root  of  which  is  5,  the  diameter  in  inches  of  the 
east  iron  gudgeon  :  then  say. 

As  14  :  125::  9  :  80-357. 
The  cube  root  of  which  is  4-30887*. 

Upon  this  principle  the  following  table  is  calculated  to 
Aaw  the  proportionate  diameters  of  cast  iron  and  wrought 
iran  gudgeons. 

nPLARATION  OF  THE  TIBLB  OF  CAST   AND  WROUGHT   IRON   OUDOE0N8. 

Columns  1  and  2  are  the  same  as  those  in  the  table  of 
cast  iron  gudgeons. 

Column  3  contains  numbers  in  the  proportion  of  9  to  14 
lesg  than  those  of  column  2. 

Column  4  contains  the  cube  root  of  column  3,  or  the 
diameters  of  wrought  iron  gudgeons,  having  the  same 
ttrength  as  those  of  cast  iron  in  column  1. 

*  To  find  the  proportion  by  the  preceding  note,  mnldply  the  diameter  of 
e  cut  iron  gudgeon  by  0*963.  Thus  5  inches  X  0*963  =  4*815  inches, 
e  diameter  for  a  wrought  iron  one. 


ON  THE  ■HAVTS   OF  MILU. 


TABLE  OF  CAST  AND  WBODGBT  IBON   G17DOBON8. 


1 

2 

3 

4 

Dismeterof 

Cube  of  diuneter  of 
casl  iron  gudKeoDS,  or 
the  cwte.  wEid.  the 

Cube  or  iSaineteT 

DiamMeiof 

ntnughliron 

nf 

of  wrought  iroD 

Qri,- 

gudgeon*  may  Bustain. 

[-III. 

)■ 

r 

■6428571 

-863 

1-25 

1-953125 

1-2555803 

1-063340 

1-5 

3-375 

2-1696427 

1-259921 

1-75 

5-359875 

3-4453125 

1-514825 

2- 

8- 

51428571 

1-709976 

S-SB 

11-400625 

7-3289732 

!-»12983* 

is 

15-625 

10-0446428 

2154435 

S-75 

20-798875 

13-3694196 

2-361335 

3- 

27- 

17-3571428 

2-571282 

3-25 

34-328125 

22-0670803 

2-802039 

3-5 

42-875 

27-5625 

3-018294t 

3-75 

52-734375 

33-9006696 

3-239618 

*■ 

64- 

41-1428571 

3-448217 

4-25 

76-785625 

49-3493303 

a-esgsot 

4-5 

91-125 

58-5803571 

3-881938 

4-75 

107-171875 

88-896 

4-10I5«6 

6- 

125- 

80-357 

4'808870 

S-26 

144-763125 

93023 

4-530655 

5-» 

iee-375 

106-955 

4-747459 

5-7S 

190109375 

122-213 

4-959675 

e- 

216- 

138-857 

5-180101 

e-25 

244.-1 40625 

156-948 

5-394690 

e-5 

274625 

176-545 

5-609376 

e-75 

307-546875 

197-709 

5-828476 

7- 

343- 

220-500 

6-041877 

7-26 

381-078125 

244-979 

6-257324 

7-i 

421-875 

271-205 

6-471274 

7-75 

465-484375 

299-240 

6-686882 

8- 

512- 

329143 

6-903436 

8-25 

561-515625 

360-975 

7-120367 

S-5 

614-125 

394-795 

7-337234 

8-75 

ee9-921875 

430-664 

7-553688 

9- 

729- 

468-643 

7-769462 

8-85 

791-453125 

508-791 

7-984344 

9-5 

875-375 

562-741 

8-257263 

8-75 

926-859375 

593-837 

8-415541 

10- 

1000- 

642-857 

8-631103 

10-25 

1076-890625 

692-287 

8-845085 

10-5 

1157625 

744-187 

9061309 

10-75 

1242-396873 

798-619 

9-279308 

II- 

1452- 

933-428 

9-771484 

*  Bnw)ier  aj«,  "  A  gudgwn  2  inches  disnieter,  wiO  tatoin  9S39  Dm.  noi 
BirnMhT's  Edition  of  FereuiMHi^  Leaim*.  Vol  I.  p.  157. 

f  The  wRiugfaliniD  ^ndtlonc  tpindle  ui«db}-  Snuthon,  iahk  esperimenKon 
■u31iurheii]iaii]eler,indcuTiedanDDe«>e^iiDg3TU0Il«.  Sae  FUL  Hi^  VoL  X 


I 


CHAP.m.]  ON   THE   SHAFTS   OF   MILLS.  Sll 


USE  OP  THE  TABLE, 


EXAMPLE. 


^.  To  find  the  diameter  of  a  wrought  iron  gudgeon  of 
the  same  strength  with  one  of  cast  iron  of  3  inches  diameter. 
Look  in  the  1st  column  for  S,  and  on  the  same  line  in  the 
4th  column  will  be  found  2*571  S82,  that  is,  a  little  more 
than  ijj  inches,  the  diameter  required  of  the  wrought  iron 
gudgecm. 

The  numbers  in  the  Srd  column,  being  the  cube  of  those 
in  the  4th,  another  use  may  be  made  of  this  part  of  the 
table.  For,  supposing  the  4th  column  to  represent  cast 
iron  gudgeons,  then  the  Srd  column  will  represent  the  hun- 
dredweights which  cast  iron  gudgeons  of  those  diameters 
should  sustain. 

Before  proceeding  to  consider  the  bodies  of  shafts  sub- 
ject to  lateral  stress^  we  shall  inquire  into  the  strength  of 
joumak  of  shafts  subject  to  torsion. 


CHAPTER  IV. 


X  L 


7   JIC2XALS.   WBXS  THB  STRESS  ARISES  FBOH 

a  TwrasnjKk,  is  Aoxnox  to  lateral  stress^ 


Strength  and  Durability  of 

if  Wsfesefe^  we  luiTe  used  what  is  called  the  hmsei 

h  Jt^sasfsre  iar  the  strain.     We  refer  the  reader 

-D  ^nac  iPB?  lan?  diere  said  (Art  108 — 114)  in  explana- 

imL  IT  :3ac  vm  vhieh  ve  shall  use  here,  in  measuring  the 

jriutfOi:  CB  shafts  bv  torsion  or  twisting. 

c^  torsioii,  as  well  as  that  of  lateral  pressure, 
prnpMtionate  strength  is  as  the  cubes  of  the 

It  3CIT  be  proper  here  to  remark,  that  what  we  had  to 
g  journals,  relates  to  those  of  cast  iron,  for 
iron  will  bear  more  lateral  stress,  as  we 
^Art.  *25,)  yet  it  is  a  fact,  perhaps  not  gene- 
nZy  known,  that  wrought  iron  will  not  resist  torsion  equal 
)£^  cast  iron^. 

In  seme  cases  a  journal  has  not  only  torsion  to  resisf, 
bet  abo  to  carry  a  very  heavy  fly  wheel,  and  it  is  prudent 

*  When  ft  shaft  has  a  support  between  the  points  where  the  poirer  aofi 
resuSftDce  are  applied,  the  part  of  the  shaft  which  revolves  on  this  sopport 
}<  oued  a  Journal,  i 

+  See  Gr^ry's  Mechanics,  Vol.  I.  Aft.  191. 


This  reference  is  to  a  statement  that  the  strength  is  i  awpcrtw^ 

bot  it  is  not  demonstrated. 
t  The  author  seems  to  be  under  a  miatelt^  k 


CHAP.  IV.3  ON   THE    SHAFTS   OF   MILLS.  213 

in  such  cases  not  merely  to  make  an  allowance  for  the 
weight  properly  balanced,  but  also  for  any  inaccuracy  which 
may  occasion  swagging,  which  greatly  adds  to  the  stress : 
but  others  have  hardly  any  other  resistance  but  what  arises 
fitmi  torsion. 

It  is  further  observable  that  the  value  for  10  horses  in 
the  smaller  engines  is  much  less  than  in  the  larger.  For 
this  difference  what  we  have  said  respecting  the  weight  of 
the  fly  in  a  great  measure  accounts :  and  not  only  is  the 
heavier  fly  to  be  considered,  but  also  the  greater  danger  of 
accidents  from  a  large  fly  than  from  one  that  is  of  a 
smaller  diameter. 

SECTION  11. 

OF   PBOPOBTIONINO  JOURNALS  TO  THE  STRESS  WHICH   THB7   HIVB   TO 

SUSTAIN. 

229.  The  stress  any  journal  has  to  sustain  being  as  the 
horses'  power  to  which  the  resistance  is  equal  directly ^  and 
the  number  of  revolutions  which  the  shaft  makes  inversely^ 
it  follows:  That  a  resistance  for  example  of  32  horses' 
power  on  a  journal  making  50  revolutions  per  minute,  has 
the  very  same  stress  with  a  resistance  of  16  horses'  power 
on  another  journal  making  25  revolutions  per  minute. 

dS  divided  by  50  is  equal  to  16  divided  by  25,  each  of 
which  gives  a  quotient  of  0*64«.  Therefore  in  all  cases 
when  the  horses'  power  divided  by  the  revolutions  per 
minute  produces  the  same  quotient  the  stress  is  the  same. 

Thus  a  resistance  equal  to  50  horses'  power  making  50 
revolutions  per  minute,  produces  the  very  same  stress  as  10 
horses'  power  making  10  revolutions  per  minute. 

Having  therefore  fixed  on  any  journal  which  has  been 
found  sufficiently  strong,  we  may  make  any  other  to  have 
Ae  same  strength  in  proportion  to  the  resistance  which  it 
has  to  overcome  in  the  following  manner. 


214  ON  THE   SHAFTS   OF   BflLIA  |^CBAF.  IT. 

230.  Rule. — If  it  so  happen  that  the  hone^  pom, 
and  the  revolutions  per  minute  he  the  same  mumberm  Fcr 
instance,  50  horses'  power  making  50  revolutioiiBy  5O-r50 
=  1 ;  then  the  cube  of  the  diameter  of  the  jofunal  will  h 
a  multiplier,  by  which  to  find  the  cube  of  the  diameter  of 
the  required  journal. 

But  in  case  the  horses'  power  and  the  revolatiooi  par 
minute  are  different  numbers^  then  you  must  suppose  them 
both  the  same,  and  calculate  (as  in  Ex.  II.)  what»  in  that 
case,  would  be  the  proportionate  diameter  of  the  joumaL— 
The  cube  of  this  diameter  will  be  a  multiplier,  the  saine  u 
mentioned  above. 

Having  found  the  multiplier,  to  find  the  diameter  of  the 
required  journal. 

Divide  the  horses'  power  by  the  revolutions  per  minute. 

Multiply  the  quotient  by  the  multiplier^  the  cube  root  of 
the  product  will  give  the  diameter  of  the  journal  re- 
quired*. 

EXAMPLE  I. 

To  find  the  multiplier  from  a  journal  7^  inches  diameter, 
where  there  is  an  engine  of  50  horses'  power  turning  a 
shaft,  at  the  rate  of  50  revolutions  per  minute. 

Divide  the  power  by  the  revolutions,  that  is  50  divided 
by  50  is  equal  to  1 ;  the  diameter  of  the  journal  is  7i 
inches ;  the  cube  of  this  is  4^,  which  multiplied  by  1 
produces  420. 

In  this  case  it  happens  that  the  horses'  power,  and  the 
revolutions  per  minute,  are  the  same  numbeTf  therefore  we 
with  little  trouble  find  the  multiplier. 

EXAMPLE  n. 
PVom  a  journal  of  4  inches  diameter,  where  the  horses' 

*  See  Art  233. 


CHAP.  IV.3  ON    THE    SHAFTS   OP   MILLS.  215 


is  1  ii,  and  the  revolutioDs  per  minute  4>8  ;  to  find 
c  multiplier. 

Now  let  us  suppose  both  numbers  the  same,  that  ie,  12 
irses*  power,  and  12  revolutions. 

Here  it  is  evident  that  there  will  be  four  times  the 
brought  on  the  journal.     Its  actual  diameter  was  4 


Then  the  cube  of  4  is  64, 

C-t  multiplied  by  4  is  256  inches, 

The  cube  root  of  which  is  6.35, 
riiieli  is  the  diameter  which  the  journal  ought  to  have,  to 
be  in  proportion  to  the  velocity. 

The  cube  of  6'35  is  26, 
iliich  is  the  multiplier  required. 

11  is  to  be  observed,  that  in  the  latter  case  a  much 
nailer  steam  engine  is  employed  than  in  the  former,  and, 
lerefore,    for  reasons  already  given,  (Art.  228,)  has  less 

i  brought  upon  the  joumaL     This  accounts  for  the 
ultiplier  being  less. 

We  shall  now  give  an  example  of  the  application  of  a  inul- 
Her;  let  us  take  that  found  in  the  ease  Example  I.,  viz. 
80,  and  see  what  size  of  the  journal  it  would  give  in  the 
Be  Example  II.,  which  is  an  engine  of  12  horses'  power 
id  journal  making  48  revolutions  per  minute. 

12  divided  by  48,  equal  to  -2.5,  then  multiplied  by  420, 
B  a  quotient  of  105,  that  is,  the  strength  of  the  journals 
it  be  as  420  to  105;  but  the  cube  root  of  420  is  7^, 

dthe  cube  root  of  105  is  4f,  which  points  out  that  the 
TOnials  74  and  4|  arc  proportioned  to  their  respective 
In  like  manner  the  following  table  is  calculated; 
Hie  multiplier  being  420. 

fSl.  Description  of  the  Table  of' Journals,  proportionate 
to  D,  havin-g  420  ns  a  multiplier. 
Column  1  contains  letters  to  distinguish  the  cases  in 


216 


ON  THE  SHAFTS  OF  MILLS.      [CHAP.  1?. 


which  D  is  the  some  as  Example  L  (Art.  230,)  andE, 
Example  II.  of  the  same  Art. 

Column  2  the  horses'  power. 

Column  3  the  revolutions  of  the  journal  per  minute. 

Column  4  the  product  of  the  horses'  power  divided  by 
the  revolutions  of  the  shaft. 

Column  5  contams  the  proportionate  strain  on  each 
journal,  represented  in  whole  numbers,  which  are  found  by 
multiplying  the  product  in  column  4,  by  420,  as  a  molti- 
plier. 

Column  6  diameters  of  journals,  as  really  executed  in 
several  steam  engines. 


TABLE   OF   JOURNALS 


Proportionate  to  d,  having  420  as  a  multiplier. 


1 

A 
B 
D 
E 
F 

2 

3 

4 

5 

6 

Niimhers  of 
horses*  power. 

Revolutions 

of  journal  per 

minute. 

Product  of 
power  divided 
DV  the  rev.  of 

the  journal. 

Proportionate 
strain  on 
journal. 

Diameters  of 

journals  from 

observation. 

32 
32 
50 
12 
9 

58 
19 
50 
48 
55 

0-55 

1-67 

1-0 

0-25 

0-16 

231 
701 
420 
105 
67 

9; 

4 

4 

OBSERVATIONS. 

232.  We  have  ab-eady  observed,  (Art.  228,)  that,  be- 
sides torsion,  the  journals  of  fly-wheel  shafts  have  consider- 
able lateral  and  other  stress,  arising  from  the  weight  and 
swagging  of  their  fly  wheels,  and  therefore  they  ought  to 
be  made  stronger  than  shafts,  in  other  situations.  The 
multiplier  420,  therefore,  which  we  have  used  in  the  table. 


IV.]  ON    THE    SHAFTS    OF    MILLS.  217 

eld  give  diameters  too  great,  for  some  other  parts  of 
ihinerj'.  A  journal,  for  instance,  subject  to  torsion, 
mediately  connected  with  a  water-wheel,  has,  from  the 
[ht  of  the  wheel  and  other  causes,  considerable  lateral 
» ;  but  not  80  much  as  that  of  a  steam  engine.  The 
Bnal  may,  therefore,  be  considerably  smaller  than  would 
required  for  a  steam  engine  fly-wheel  shaft  subject  to 
same  degree  of  torsion*. 

Again,  a  secondary  shaft  driven  from  a  steam  engine,  a 
T-wheel,  or  horse-gin,  by  means  of  wheels,  has  in  gene- 
very  little  lateral  stress,  compared  with  the  two  cases 
stated  ;  and  may  therefore  have  a  journal  smaller  than 
er,  when  the  degree  of  torsion  is  the  same. 
SS3.  For  these  reasons  the  three  following  multipliers 
U  probably  approach  near  the  truth ;  that  is,  for  journals 
steam  engine  fly-wheel  shafts  (where  the  power  is  mo- 

ite) 400 

Journals  in  immediate  connexion    with   water-wheels  t, 

other  heavy  work 300 

Journals  for  the  ordinar>'  kind  of  internal  mill-work  100 


Suppo8e  B,  in  the  table,  (Art.  231,)  1-67  multiplied  by 
0,  is  668,  the  cube  root  of  which  is  8'74l6  inches,  dia- 

of  journal. 

SS*,  When  the  resistance  of  a  journal  is  equal  to  the 
'ittbg  stress,  the  strain  not  being  sufficient  to  produce 
nnanent  derangement  in  the  material,  the  cube  of  the 
meter  of  the  journal  will  be  equal  to  3*78  times  the 

It  wems  B  better  method  to  use  a  mode  of  c&Iculadon  which  includes 
dectof  lateToI  streBs;  see  Art.  333. 

The  reader  will  jtletme  to  obaerre,  that,  when  Bnchamui  uses  the  word 
ol  here,  he  suppows  it  subject  to  torsion.  Where  there  is  latenl 
KK  only,  and  no  tornon,  he  invariably  uses  the  word  gudgeon. 


218  ON  THE   SHAFTS   OF   MILLS.  [CHAP.  IT. 

number  of  horses*  power  divided  by  the  number  of  revo- 
lutions in  a  second*. 

But  some  allowance  must  be  made  for  wear,  and  if  this 
be  made  so  that  the  journal  shall  have  sufficient  strength 
when  it  is  worn  down  one-sixth  of  its  diameter,  the  nonher 
8*78  should  be  made  6*01 ;  or  with  sufficient  accuracy  & 

If  the  number  of  revolutions  in  a  minute  be  employed 
instead  of  those  in  a  second,  the  constant  multiplier,  6| 
must  be  multiplied  by  60 ;  and  therefore  the  constant  mul- 
tiplier will  become  360 ;  and  a  less  number  ought  not  in 
any  case  to  be  employed,  because  there  will  always  be  some 
lateral  stress  in  addition  to  the  twisting  stress.  The  re- 
sistance of  a  journal,  or  its  diameter  as  regards  the  twist- 
ing strain,  may  be  always  calculated  by  the  following  rules. 

Rule. — If  n  be  the  number  of  revolutions  in  a  minute, 

N.d? 
and  d  the  diameter  of  the  journal  in  inches,  then  Qkf;=thfi 

number  of  horses'  power  the  journal  is  sufficient  to  resist. 
Rule. — If  n  be  the  number  of  revolutions  per  minute, 
and  H  the  number  of  horses'  power  moving  the  train  of 

machinery,  then  7*12  x  (— )  =  rf,  the  diameter  of  the  jour- 
nal in  inches. 

EXAMPLE. 

Let  it  be  required  to  find  the  diameter  of  a  journal  for 

case  B  in  the  table  of  journals,  Art.  231. ;  then  we  have  h 

equal  32  horses,  and  the  number  of  revolutions  n  equal  19; 

32 
therefore  Yq  =  1*68421.      The   cube  root  of  1-68421   is 

found  to  be  1-19;  and  7*12  x  1-19  =  8-4728  inches,  the 
diameter  of  the  journal,  or  nearly  8^  inches. 

*  The  reaaon  of  this  rule  will  be  given  in  treating  of  the  resistance  d 
shafU. 


CHAP.  IV.]  ON   THE    SHAFTS   OF    MILLS.  219 

235.  But  the  effect  of  lateral  stress  ought  always  to  be 
considered,  and  we  have  found  the  strength  of  a  gudgeon 
to  be  0"6(w/)i  =  rf;  where  the  stress  is  wholly  lateral, 
(Art  212.)  Or,  0-216  w/  =  flP.  And  the  strength  of  a 
journal,   where  the  stress  is  altogether  twisting,   to  be 

=  flP ;   therefore,  since  journals  bear  both  kinds  of 

stress,  we  have  as  a  general 

Rule.  —  (0'2l6  w/  -f )    =  rf,  the   diameter  in 

inches.  Where  w  is  the  lateral  stress  upon  the  journal  in 
cwts.,  /  the  length  of  the  journal  in  inches,  h  the  number 
of  horses'  power  moving  the  train  of  machinery,  and  n  the 
number  of  revolutions  of  the  journal  per  minute. 


SECTION  III. 

236.  When  the  diameter  of  a  journal  and  its  revolutions 
per  minute,  are  given,  in  order  to  find  the  horsed  power  to 
which  it  is  equal ;  we  must  invert  the  preceding  operation, 
and  convert  the  multiplier  into  a  divisor. 

Rule. — Cube  the  diameter  of  the  journal,  divide  the 
cube  by  the  divisor.  The  quotient  multiplied  by  the  re- 
volutions per  minute,  gives  the  horses'  power,  to  which 
the  journal  is  equal . 


EXAMPLE   I.— F   IN   THE  TABLE. 

Suppose  the  journal  of  a  steam  engine  to  be  4  inches 
diameter,  making  55  revolutions  per  minute,  then  we  shall 
use  400  as  a  divisor. 

The  cube  of  4  is  64,  divided  by  400  equal  to  0*1 6,  then 
this  quotient  (0*16)  multiplied  by  55,  gives  8*8  the  horses' 
power,  to  which  the  journal  is  equal 


9S0  ON  THE   SHAFTS   OF   MILLS.  [CHAP.  T 


EXAMPLE  n. 


Suppose  again,  the  same  size  of  a  journal,  connecti 
with  heavy  machinery,  then  we  must  use  200  as  a  diviso: 

The  cube  of  4  =  64  -r-  200  =  -32  x  55  =  lyG  horsi 
power. 


EXAMPLE  m. 


We  shall  take  the  same  journal  for  internal  work,  of  tl 
ordinary  kind,  and  use  100  as  a  divisor. 

The  cube  of  4  =  64  -r  100  =  -64  x  55  =  35-2  horsa 
power. 


CHAPTER  V. 


SECTION  I. 


ON   THE   BODIES  OF   SHAFTS. 


237.  From  what  is  stated  in  the  preceding  part  of  this 
Essay,  the  millwright  will  he  enahled  to  approach  suffi- 
ciently near  to  the  truth,  for  all  practical  purposes,  in  pro- 
portioning gudgeons  and  journals  to  the  stress  which  they 
have  to  sustain.  Taking  this  for  granted,  it  may  he  pro- 
per for  us  next  to  consider  what  relates  to  the  bodies  of 
shafts,  or  those  parts  which  lie  hetween  the  gudgeons  or 
journals.  In  this  part  of  our  inquiry,  we  may  derive 
assistance  from  the  principles  which  have  heen  applied  hy 
writers  on  mechanics,  to  the  stress  of  timher  and  other 
materials*.  The  generality  of  writers  on  mechanics,  how- 
ever, as  Dr.  Young  justly  observes,  (vol.  i.  page  136,) 
have  confined  their  attention  to  strength  (resistance  to 
fracture)  alone,  although  there  be  other  very  important 
properties,  which  required  their  consideration.  The  most 
usual  as  well  as  the  most  important  effect,  produced  by  the 
application  of  force  is  flexure :  (p.  138,  ibid.)  stiffness 
therefore,  as  well  as  strength^  ought  to  be  considered  in 
determining  the  form,  as  well  as  the  quantity  of  materials, 
for  any  mechanical  purpose,  more  particularly  that  of  a 
Bhaffc  in  mill- work,  which  in  theory  may  be  considered  as 
an  inflednhle  straight  line.  The  practical  reader  should 
attend  to  the  distinction  between  stiffness  and  strength. 

*  Few  operatiTe  mechanics  hare  a  distinct  notion  of  the  difference  be- 
tween ilnngtk  and  tHfiteis. 


Q2fi  ON   THE    SHAFTS    OF   MILLS.  [CHAP.T. 

Stiffiiess  is  that  property  which  resists  ^jrt^r^  or  ben^ng. 

Strength  that  which  resists  fracture  or  hreaking. 

The  consideration  of  their  limits,  may  make  this  plainer. 

The  limit  of  stiffness  \&  flexure  ;  the  limit  of  strength  is 
Jracture. 

The  stiffness  of  a  heam  follows  laws  very  different  from 
those  which  determine  its  strength ;  these  laws  we  shall 
presently  consider,  and  endeavour  to  shew  their  application 
to  practice,  with  regard  to  some  cases  of  shafts.  But 
although  those  laws  may  throw  considerahle  light  on  the 
suhject,  yet  it  must  he  confessed,  that  there  are  many  cases 
in  practice,  in  which  it  is  very  difficult,  if  not  impossible, 
to  apply  them ;  for  it  is  very  often  difficult  to  estimate 
what  may  he  the  amount  of  the  lateral  pressure  on  shafts, 
arising  not  only  from  their  own  weight  and  that  of  the 
wheels,  upon  them,  hut  also  from  the  thrust,  proceeding 
from  the  action  of  the  toothed  wheels,  and  other  extnu 
neous  causes.  In  cases  of  this  nature,  where  calculation 
fails,  much  must  he  done,  hy  what  Smeaton  calls  feeling*^ 
which  will  direct  the  experienced  millwright  to  make  a  due 
allowance  for  whatever  accidental  strain  may  be  Ukelv  to 
occur. 

We  shall  now  proceed  to  state  and  apply  some  of  the 
laws,  respecting  stiffness  and  strength,  with  regard  to 
force,  applied  transversely. 


OF   LATERAL   STIFFNESS   AND   LATERAL   STRENGTH. 

238.  The  "  stiffness  of  any  substance,  is  measured  by  the 
force  required  to  cause  it  to  recede,  through  a  given  small 
space,  in  the  direction  of  the  force,"  (Young's  Nat.  Phil- 
vol.  i.  p.  139.)  Its  transverse  strength  is  measured  by  the 
pressure  required  to  produce  its  fracture,  or,  in  other 
words,  to  break  it. 

*  Smeaton  8  Account  of  Eddjstone  Lighthouse,  p.  136. 


OX    THE    SHAFTS    Of    MILLS. 


TROPOSITION  11. 


Any  beams  of'  equal  length  have  tfieir  lateral 
'M,  [/o  bear  a  load  at  any  point  in  t/ie  length,']  as 
breadth  and  cube  of  the  depths  (Young's  Nat.  Phil, 
vol  i.  p.  139,  or  ii.  art.  333,)  and  have  their  lateral 
ttretigth,  atf  the  breadth  and  square  of  Die  depth*,  (Gre- 
vol.  i.  art.  169,  cor.  1.  Emerson,  prop.  67.) 
IU3,  if  a  square  beam  measure  twice  as  mucli,  on  the 
le,  as  another  of  equal  length,  it  would  be  sixteen 
times  as  stiff.  In  other  words,  it  will  sustain  sixteen  times 
le  weight,  without  bending. 

But,  if  a  square  beam  be  twice  aa  much  on  the  side  as 
ither,  both  being  the  same  length,  it  will  be  only  eight 
B  stronger. 

!ence  we  see,  that  when  beams  or  shafts  are  of  equal 
ths,  their  stiffness,  by  any  increase  of  thickness,  in- 
Bes  in  a  higher  proportion  than  their  strength. 


EXAMPLE  1. 

f  a  beam  or  shaft  be  four  inches  square  throughout, 

another  five  inches,  both  of  equal  lengths ;  what  is 

■  comparative  stiffness  f 

The  cube  of  %  is  6+, 

(Mix  4  =256, 

The  cube  of  5,  is  1^25, 

125  X. 5  =625, 

That  is,  the  shaft  of  live  inches  is  nearly  two  and  a  half 
stiffer  than  that  of  four;  in  other  words,  it  would 
[lire  nearly  two  and  a  half  times  the  weight  to  Iwud  it. 

Ihat  h,  as  tlie  cube  of  the  «de  of  a  square  beam,  sjid  in  general  tlic 

Kof  any  beams  wbose  sections  are  similar,  as  the  cube  of  ibc 
diametere  of  tbc  seclioiis. 


224  ON   THE    SHAFTS   OF   MILLS.  [CHAP.  V. 

EXAMPLE  II. 

If  a  beam  or  shaft  be  four  inches  square  throughout,  and 
another  five  inches,  both  of  equal  lengths ;  what  is  their 
comparative  strength  ? 
The  cube  of  4,  is  64, 
The  cube  of  5,  is  125. 
That  is,  the  five  inches  shaft  is  nearly  twice  as  strong  as 
that  of  four  inches ;  in  other  words,  it  would  require  nearly 
double  the  weight  to  break  it. 


PROPOSITION  III. 

240.  Any  beams  of  different  lengths  have  their  stiffness 
[to  bear  a  load  at  any  point  in  the  length^  directly  as  the 
breadth  and  the  cube  of  the  depth,  and  inversely  as  the 
cube  of  the  length,  (Young^s  Nat.  Phil.  ii.  art.  333,)  and 
have  their  strength  directly  a^  the  breadth,  and  as  the 
square  of  the  depth,  and  inversely  as  the  length*,  (Young, 
vol.  ii.  art.  335.) 

Thus,  if  a  beam  be  twice  as  long  as  another,  of  the 
same  breadth  and  depth,  it  will  have  only  one  eighth  of 
the  stiffness,  while  it  will  have  one  half  of  the  strength. 
Hence  the  stiffness  of  shafts  or  beams  by  any  increase  of 

•  This  is  not  strictly  true  in  practice,  for  "  some  experiments  appear  to 
shew,  that  the  strength  is  diminished,  in  a  proportion  somewhat  greater 
than  that  in  which  the  length  is  increased."  (Young's  Nat.  Phil.  vol.  ii.  p* 
147.) 

The  variation  is  caused  by  the  increase  of  strain  which  takes  place  wbc^ 
the  flexure  is  considerable,  (see  Elementary  Principles  of  Carpentry,  Art 
18,)  and  some  decrease  of  cohesive  power  when  the  natural  arrangement 
of  the  particles  of  a  body  is  disturbed  more  than  in  a  certain  degree;  but 
these  causes  are  insensible  in  a  practical  point  of  view,  because  we  can 
never  allow  the  stress  to  produce  so  much  flexure,  nor  die  strain  to  be  00 
near  to  fracture,  as  to  make  it  necessary  to  allow  for  such  circumstances. 


CHAP,  v.]  ON   THE    SHAFTS   OF   MILLS.  225 

their  length,  decrease  in  a  much  higher  proportion  than 
that  of  their  strength. 


EXAMPLE  I. 

Suppose  a  beam  or  shaft,  four  feet  long  and  four  inches 
square  throughout,  and  another  eight  feet  long  and  seven 
inches  square ;  what  is  their  comparative  stifihess  ? 
The  cube  of  4  feet,  is  64, 
The  cube  of  8  feet,  is  512, 
512  divided  by  64,  is  equal  to  8,  that  is,  when  we  double 
the  length,  we  decrease  the  stifihess  eight  times. 

The  cube  of  4  inches  is  64,  which  multiplied  by  4  is 
equal  to  256,  a  number  representing  the  stifihess  of  the 
four  inch  shaft. 

The  cube  of  7  inches  is  343,  multiplied  by  7  is  equal  to 
£401,  divided  by  8  is  equal  to  300*1,  then  as  256  is  to 
300*1,  so  is  the  stifihess  of  the  shaft  of  four  inches  to  that 
of  a  shaft  of  seven  inches. 


EXAMPLE  n. 

Suppose  a  beam  or  shaft,^  four  feet  long  and  three  inches 
square,  and  another  eight  feet  long  and  seven  inches 
square,  what  is  their  comparative  strength  ? 

The  cube  of  4  is  64,  which  represents  the  strength  of 
the  four  inch  shaft. 

The  cube  of  7  is  343 ;  but  the  shaft  being  of  double 
length,  we  must  halve  this  sum,  to  find  the  number  repre- 
senting its  strength,  viz.  343-^2  =  171*5  divided  by  100, 
that  is,  as  sixty  four  is  to  a  hundred  and  seventy-one  and 
a  hal^  80  is  the  strength  of  the  short  shaft  to  that  of  the 
long  one.  Thus  the  shaft  of  seven  inches,  eight  feet  long, 
has  nearly  two  and  six  tenths  times  the  strength,  of  the 
firar  inch  four  feet  long. 


2^  ON    THE   SHAFTS   OF   MILLS.  [CHAP.  V. 

PROPOSITION  IV. 

241.  Supposing  a  tubcy  indefinitely  thin^  to  be  expanded 
into  a  similar  tube  of  a  greater  diameter^  but  of  equd 
lengthsy  the  quantity  of  matter  remaining  the  same^  tiie 
STIFFNESS  will  be  increa^edy  in  the  ratio  *  of  the  square  of 
the  diameter^  and  the  strength  in  the  ratio  of  the  dia- 
meter f. 

Thus,  if  the  one  tube  be  double  the  diameter  of  the 
other,  it  will  have  four  times  its  stiffness^  but  only  double 
the  strength. 

Hence,  hollow  cylinders  of  equal  lengths,  by  any  in- 
.crease  of  diameter,  increase  in  stiffness^  in  a  much  higher 
proportion  than  in  strength. 

EXAMPLE  I. 

Suppose  two  thin  narrow  cylindrical  cast  iron  shafts,  of 
equal  lengths  and  weights,  the  one  of  one  foot  diameter, 
and  the  other  three  feet  diameter,  required  their  compara- 
tive stiffness  ? 

The  square  of  1  is  1, 

The  square  of  3  is  9, 
that  is,  the  shaft  of  three  feet  diameter,  is  nine  times  stiffer 
than  that  of  one  foot. 

*  Ratio,  that  is,  proportion. 

t  This  proposition  is  taken  from  Dr.  Young's  Nat.  Phil.  vol.  ii.  wl.  339, 
where  it  is  followed  by  this  essential  limitation.  "  When  a  beam  of  finite 
thickness  is  made  hollow,  retaining  the  same  quantity  of  matter,  the 
strength  is  increased  in  a  ratio  somewhat  greater  than  that  of  the  diameter, 
because  the  tension  of  the  internal  fibres  at  the  instant  of  breaking  is  m- 
creased."  Dr.  Young  has  given  the  correct  rule  for  estimating  the  strengtb 
and  stiffness  of  a  hollow  cyHnder,  at  p.  84,  (Nat.  Phil.  vol.  ii.)  "  The 
strength  of  a  tube  may  be  found  by  deducting  from  the  strength  of  the 
whole  cylinder  that  of  the  part  removed,  reduced  in  the  ratio  of  tlie  dia- 
meters." And  observes,  that  "  the  strength  is  in  this  case  in  the  sune 
ratio  as  the  stiffness." 


CHAP,  v.]  ON   THE   SHAFTS   OF   MILLS.  227 

EXAMPLE  II. 

Suppose  the  same  shafts,  as  in  example  first,  required 
their  comparative  strength  ? 

Diameter  one  foot, 

Diameter  three  feet, 
that  is,  the  three  feet  shaft  is  just  three  times  stronger 
than  that  of  one  foot  diameter. 

242.  In  these  examples,  we  have  supposed  the  weight  of 
the  shafts  equal,  that  is,  the  area  of  their  ends  to  be  equal, 
but  the  strength  of  any  of  them  would  be  increased  in 
proportion  to  their  weight,  or  the  areas  of  their  ends  and 
diameters,  conjointly.     (Gregory,  vol.  i.  art.  172,  cor.  3.) 

Thus,  suppose  two  shafts  of  equal  length  and  diameter, 
the  one  double  the  weight  of  the  other,  it  will  be  double 
the  strength*. 

243.  Professor  Robison  justly  observes,  "  that  this 
property  of  hollow  tubes  is  accompanied  also  with  greater 
stifihess,  and  the  superiority  in  strength  and  stifiness  is  so 
much  the  greater,  as  the  surrounding  shell  is  thinner  in 
proportion  to  its  diameter.  Here  we  see  the  admirable 
wisdom  of  the  Author  of  nature  in  forming  the  bones  of 
animal  limbs  hollow.  The  bones  of  the  arms  and  legs 
have  to  perform  the  office  of  levers,  and  are  thus  opposed 
to  very  great  transverse  strains.  By  this  form  they  be- 
come incomparably  stronger  and  stiffer,  and  give  more 
room  for  the  insertion  of  muscles,  while  they  are  lighter 
and  therefore  more  agile ;  and  the  same  wisdom  has  made 
use  of  this  hollow  for  other  valuable  purposes  of  the  ani- 
mal economy.  In  like  manner,  the  quills  in  the  wings  of 
birds  acquire  by  their  thinness  the  very  great  strength 
which  is  necessary,  while  they  are  so  light  as  to  give  suffi- 
cient buoyancy  to  the  animal,  in  the  rare  medium  in  which 

*  Sec  note  to  Art.  236. 


SS8  ON   THE   SHAFTS   OF   MILLS.  [cHAP.  V. 

it  must  live  and  fly  about  The  stalks  of  many  plants, 
such  as  all  the  grasses,  and  many  reeds,  are  in  like  man- 
ner hollow,  and  thus  possess  an  extraordinary  strength.'' 
(Ency.  Brit,  article  Strength.) 

•  Long  before  this  eminent  philosopher,  the  celebrated  Ga- 
lileo made  similar  observations,  and  goes  on  to  say  that  **if 
a  wheat  straw,  which  supports  an  ear  that  is  heavier  than 
the' whole  stalk,  were  made  of  the  same  quantity  of  mat- 
ter but  solid,  it  would  bend  or  break  with  far  greater  ease 
than  it  now  does.  And  with  the  same  reason  art  has  ob- 
served and  experience  confirmed,  that  a  hollow  cane  or 
tube  of  wood  or  metal,  is  much  stronger  and  more  finn 
than  if,  while  it  continued  of  the  same  weight  and  length, 
it  were  solid,  as  it  would  then  of  consequence  be  not  so 
thick. 

It  may  be  proper  now  to  consider  the  effects  called 
stressj  which  are  produced  on  beams  or  shafts  lying  hori- 
zontally by  weights  or  pressures  brought  on  various  parts 
of  them. 

SECTION  II. 

OP   LATERAL   STRESS. 

244.  The  stress  or  strain  *  are  terms  used  to  express 
the  force  which  is  excited  in  any  body  tending  to  break 
it  The  meaning  of  the  term  stress  may  perhaps  be  more 
clearly  understood  by  contrasting  it  with  the  term 
strength. 

Strength^  as  we  have  already  observed,  is  the  property 
which  resists  fracture. 

Stress  is  that  which  has  the  tendency  to  produce  Jrac- 
ture  ;  and  lateral  stress  is  that  particular  application  of  it, 
which  has  the  tendency  to  break  a  body  across. 

*  Strain  is  the  effect  of  stress :  it  is  the  derangement  from  the  uatnral 
state  which  is  caused  hv  stress. 


ON    THE    SHAFTS    OF    MILLS. 


^9 


PROPOSITION  V. 


77(«  stress  on  a  beam  arising  from  one  weight 
upon  it,  is  proportional  to  the  rectangle  of  the  parts 
the  beamy  and  is  greatest  when  the  load  is  laid  on  the 

of  the  beam.     (Ency.  Brit.  art.  Roof,  §  19.) 
What  is  meant  by  the  expression  rectangle  of  the  parts, 
is  the  product  of  parts  multiplied  into  each  other.     Thus, 
fcr  example ;  if  a  beam  be  ten  feet  long,  and  the  weight 
Dg  two  feet  from  one  end,  the  parts  are  2  and  8,  which 
UtipUed  together,  would  be  equal  to  KJ;  but  supposing 
iveight  were  hung  in  the  middle,  the  parts  are  5  and  5, 
ich  multiplied  together  would  produce  2.5. 
06.  The  ends  of  beams  having  the  whole  weight  to  sup- 
rt,  the  end  which  is  nearest  the  weight  has  to  support  the 
utest  proportion  of  it,  in  the  inverse  proportion  of  the 
(liatance  of  the  weight  from  the  end.     This  will  be  easily 
understood  from  the  properties  of  the  lever.     For,  suppose 
As  beam  instead  of  being  supported  by  two  props  or  walls, 
in  Fig.  9.  No.  1,  to  be  hung  from  each  end  by  a  rope, 
b  Fig.  9,  No.  %  it  is  plain  that  the  beam  would  receive 
Bame  support,  and  suffer  the  same  stress,  as  if  Iving  on 
pe  or  walls ;  now  suppose  the  weights  a  and  b,  to  ba- 
le the  weight  w,  then  a  and  b,  taken  together,  must  be 
ll  to  w,  but  A  must  be  greater  than  b  in  proportion  as 
1  near  to  it. 

St?"  Hence  when  any  beams  or  shafts  are  loaded  exactly 
he  middle,  each  of  the  ends  of  the  beams  or  gudgeons 
the  shafts  has  half  the  weight  to  support,  and  when  the 
ight  is  nearer  one  end,  the  end  or  gudgeon  to  which  it 
tearestt  has  the  stress  in  the  inverse  proportion  of  the 
In  this  last  case,  therefore,  the  one  gudgeon 
^t  be  smaller  than  the  other. 
(S.  "  We  may  (Uways  consider  the  weight  which    is 


2S0  ON   THE   SHAFTS   OF   MILLS.  [CHAP.  V. 

uniformly  diffused  over  any  part  of  a  beam  as  united  in 
the  middle  of  that  part,  and  if  the  load  is  not  uniformly 
diffiised,  we  may  suppose  it  united  at  its  centre  of  gravity." 
(Ency.  Brit  article  Roof,  §  20.)* 

249.  It  is  evidently  of  importance  that  a  beam  or  shaft 
should  in  every  part  be  able  to  resist  the  strain  excited  in 
that  'psLTt.  **  It  should  therefore  be  equally  strong,  because 
the  piece  will  nevertheless  break  where  it  is  not  stronger 
throughout,  and  it  is  useless  to  make  it  stronger  (relatively 
to  its  strain)  in  any  part,  or  it  will  nevertheless  equally 
foil  in  the  part  that  is  too  weak."     (Ibid.) 

250.  From  what  we  have  said  respecting  lateral  stress, 
it  is  evident  than  when  a  beam  lying  between  two  props  is 
loaded  at  some  intermediate  part,  that  part  has  to  sastain 
more  stress  than  the  rest.  In  order  to  resist  this  strain, 
therefore,  and  to  render  the  beam  equally  strong  through- 
out, it  should  have  its  section  enlarged  at  the  place  of 
greatest  stress,  and  hence  shafts  subject  to  lateral  stress 
should  swell  in  the  middle,  and  it  will  be  found  that  whffl 
each  section  is  made  proportional  to  the  stress  it  has  to 
sustain,  that  the  sides  of  the  shaft  will  form  curves. 

251.  When  the  transverse  sections  of  a  beam  are  all 
similar,  such  as  circles,  squares,  or  polygons,  and  the 
weight  is  laid  on  one  place,  in  order  to  make  it  equally 
strong  throughout  its  length,  the  curve  of  the  sides  of  the 
beam  becomes  what  mathematicians  call  a  cubical  parabohu 
(Ency.  Brit.  Strength  of  Materials,  87.)  But  when  the 
weight  is  uniformly  diffused  all  over  the  beam,  the  sides  of 

*  When  the  weight  is  uniformly  diflfused,  the  stress  is  the  greatest  at  the 
middle  of  the  length,  and  is  equal  to  half  the  weight  collected  in  the  middk^ 
But  the  flexure  in  the  middle  produced  hy  a  weight  which  is  uniformlj 
diffused,  is  the  same  as  when  five  eighths  of  the  load  is  collected  in  the 
middle  of  the  length.  (See  art.  Carpentry,  Supplement  to  Encydopsdia 
Brit.  1817,  Prop.  F.,  or  Barlow's  Essay  on  the  Strength  of  Timher,  p. 
117.) 


CHAP,  y.3  ON   THE   SHAFTS   OF    MILLS.  231 

the  beam  must  be  a  different  parabolic  curve  called  a 
semicubical  parabola*. 

We  come  now  to  examine  some  of  the  laws  respecting 
twisting  or  torsion. 

SECTION  in. 

OF  TOBSION. 

PROPOSITION  VI. 

252.  In  general  the  strength  of  a  cylinder  or  solid  axle 
hjf  which  it  resists  being  wrenched  asunder  hy  twisting  is 
as  the  cube  of  its  diameter.  (Ency.  Brit  art.  Strength  of 
Materials,  123.) 

Thus,  if  a  solid  cylinder  be  double  the  diameter  of 
another,  it  would  require  eight  times  the  force  to  wrench 
it  asunder. 

OF  HOLLOW    AXLES. 

253.  Hollow  axles  are  stronger  to  resist  twisting  than 
solid  ones  containing  the  same  quantity  of  matter.  For  if 
a  hole  be  bored  out  of  an  axle  of  half  its  diameter,  this  re- 
duces its  weight  one-fourth^  (because  circles  are  to  one 
another  as  the  squares  of  their  diameters,)  but  the  strength 
of  solid  cylinders  being  as  the  cubes  of  their  diameters,  the 
part  taken  out  by  boring  had  only  the  eighth  part  of  the 
strength  of  the  whole  cylinder,  and  therefore  when  taken 
out  would  reduce  the  strength  of  the  whole  one  eighth. 

Thus,  let  the  external  diameter  of  the  hollow  axle  be 
fiye  inches,  and  that  of  the  hollow  part  of  it  four  inches, 
then  the  diameter  of  another  cylinder  made  solid,  having 
ih6  same  quantity  of  metal  with  the  tube,  is  three  inches. 

*  ^  Tbe  poiabola  is  a  conic  section,  arising  from  a  cone  being  cat  by  a 
phnn  panUel  to  one  of  its  sides,  or  parallel  to  a  plane  that  touches  one 
wim  of  the  eone.* 


292  ox   THE   SHAFTS   OF   MILLS.  [CHAP.T. 

For  5  multiplied  by  5  is  equal  to  25 
4  multiplied  by  4  is  equal  to         16 

Difference  9 
The  square  root  of  9  is  3.  The  strength  of  the  solid 
cylinder  of  fire  inches  diameter  may  be  expressed  by  the 
cube  of  5,  or  123.  Of  this  the  internal  part,  four  inches 
diameter,  exerts  64,  that  is,  the  cube  of  4 ;  therefore  the 
strength  of  the  tube  is  64,  subtracted  from  125,  is  equal  to 
61,  but  the  strength  of  the  solid  axle  of  the  same  quantity 
of  matter,  and  three  inches  diameter  is  expressed  b?  the 
cube  of  3  or  27,  which  is  not  half  of  that  of  the  tube. 
(Ency.  Brit  art.  Strength  of  Materials,  124.)* 

254.  The  superiority  of  strength  of  hollow  tubes  over 
solid  cylinders  is  much  greater  in  resisting  torsion  thin 
transTcrse  or  lateral  stress.  We  have  seen  above,  that  the 
strength  to  resist  torsion  of  the  tube  was  to  that  of  the 
cvlinder  as  sixtv-one  is  to  twentv-seven ;  but  Professor 
Robison  estimates,  that  their  strength  to  resist  transverse 
strain  is  onlv  as  sixtv-one  is  to  thirtv-two  and  a  half  nearly 
— and  if  we  calculate  according  to  Dr.  Gregor}^*s  corollary, 
Vol.  I.  page  109,  (see  Art.  237  of  this  Essay,)  the  result 
will  be  still  more  in  favour  of  strength  to  resist  torsion; 
for  bv  the  last  mode  of  calculation  the  tube  would  be  to  the 
cylinder  only  as  fort}-five  is  to  thirty-six ;  but  the  Pro- 
fessor's mode  of  calculation,  though  less  simple,  is  probably 
more  accurate  than  that  above  alluded  tot. 

*  These  calculations  are  founded  on  the  erroneous  supposition  that  the 

tension  is  equal  in  every  part  of  the  section ;  and  consequently  they  are 

widely  distant  from  the  truth.     (See  note  to  Art  254.) 

t  It  has  been  shewn  that  the  resistance  of  a  cvlinder  to   torsion  k 

124'8</*  =:  R  w  ;  where  w  is  the  stress  in  Ihs.  and  r  the  leverage  in  feet  it 

acts  with ;  d  being  the  diameter  of  the  cylinder  in  inches.     (Essay  on  Cast 

Iron,  Art.  227.)     And  when  the  straining  force  is  considered  to  act  at  the 

d 
surface  of  the  shaft  ^  =  1 2  r  ;  and  therefore  in  this  case  20*8  d'  zz  w. 


w.  v.] 


ON  Tiir;  sirAiTs  of  mills. 


2SS 


Profiwsor    Robison    mentions    (in     Enc;\     Brit.     art. 

;ngth  of  Materials,  §  128)  that  "when  the  matter  of 
B  axle  is  of  the  most  simple  texture,  such  as  that  of 
itak,  we  do  not  conceive  that  the  length  of  the  axle  has 
f  influence  on  the  fracture.  It  is  otherwise  if  it  be  a 
reus  texture,  like  timber ;  the  fibres  are  bent  before 
teking,  being  twisted  into  spirals  like  a  corkscrew.   The 

ph  of  the  axle  has  somewhat  of  the  influence  of  a  lever 
this  case,  and  it  is  easier  wrenched  asunder  it"  long. — 
BCordingly  we  have  found  it  so  ;  but  we  have  not  been 

e  to  reduce  this  influence  to  calculation*. 

855.  All  shafts  are  exposed  to  lateral  stress  and  twist- 

,  but  it  will  commonly  happen  that  one  of  these  forces 

'.  vastly  exceed  the  other ;  and  consequently,  we  need 

r  ad^t  the  shaft  to  the  resistance  of  the  greater  power; 

'  dte  same  reaaoning  il  may  be  proved  tliat,  in  a  hollow  cylinder,  where 
the  exterior  diameter,  and  jid  the  diameter  of  the  hollow  part, 
!tr'  (I  —  «')  =  RW  ;  wlien  the  force  w  acU  with  the  leverage  R  in 
hat  when  the  force  is  applied  at  the  surface  of  the  shaft,  20'8  n' 

when  the  strain  is  at  the  surface  of  the  shaft,  the  reaiatance  of  a 
qrlinder  is  to  that  of  a  hollow  one  as  rf'  :  »'  (1  —  »*);  and  taking  the 
I  Bumple,  wtiich  our  author  has  quoted  from  Professor  Robison,  we 
I  rf  =  3,  I)  =  5,  and  n  D  =:  4,  or  n  =  -8,  the  ratio  is  9  :  25 
-  t09e,)  or  as  9  :  14-76,  when  the  stress  is  reduced  to  the  surface  of 
liafl  or  cylinder. 

U  the  ratio  is  (/'  :  d'  ( I  —  n']  when  the  lever^e  is  constant ;  that  is, 
T3-8,  instead  of  27  to  61.  1 1  must  however  bo  observed,  that  these 
•  obt^n  only  in  tiie  particular  case  for  which  they  are  here  calculated, 
fluexample  where  the  leverage  is  constant  the  general  ratio  is  (I  —  ti-}J 
-  n')  ; :  strength  of  a  solid  cylinder  :  that  of  a  tube  containing  the 
quantity  of  matter.     And  it  may  be  very  easily  proved  that  the  lateral 

in  the  same  rabo. 
In  wood  the  lateral  adhesion  of  the  fibres  being  much  inferior  to  their 
t  cabenon,  it  b  much  ivcakcr  to  rcRist  torsion  ;  the  fibres  sliditig  one 
I  uotiier  very  considerably  before  the  rupture  takes  place.     But  this 
I  die  lenph  no  senmble  inHuence,  when  the  strain  is  kept  within  proper 


234  ON   THE    SHAFTS    OF    MILLS.  [cHAP.  V. 

but  in  the  first  place  we  must  be  able  to  ascertain  when 
the  one  or  the  other  must  be  calculated  for. 

The  resistance  of  a  shaft  to  a  lateral  stress  must  ob- 
viously be  measured  by  its  stiffiiess  to  resist  flexure,  be- 
cause it  would  otherwise  play  too  much  on  its  brasses, 
couplings,  &c,  and  occasion  irr^ular  action  in  the  maclm 

From  a  comparison  of  shafts  in  use,  it  appears,  that 
about  the  yi^th  part  of  an  inch,  for  each  foot  in  length,  is 
the  quantity  of  flexure  that  may  be  allowed  without  sen- 
sibly affecting  the  regularity  of  its  motion. 

In  a  cast  iron  shaft,  supposing  it  to  be  a  solid  cylinder, 
if  w  be  the  stress  when  referred  to  the  middle  of  the  length 
of  the  shaft ;  /  ^  the  length  in  feet,  and  d  =  the  diameter 

in  inches :  the  deflexion  in  the  middle  beinsr  of  an 

^   100 

inch,  then  ^r^  =  rf*.     (Essay  on  Cast  Iron,  Art  218.)  or 

250  rf* 
w  =  -^. 

740h 
256.  Now  if be  the  twisting  power  collected  at  the 

surface  of  the  shaft,  where  v  is  the  velocity  of  that  surface 
in  feet  per  second,  and  h  the  greatest  number  of  horses* 
power  that  is  necessary  to  work  the  train  of  machinery  to 
which  the  shaft  belongs,  (see  note  to  Art,   106,)  we  have 

740  H 
124-8cP  =^-^ — ,  (Essay  on  Cast  Iron,  Art.  227,)  hut  the 

velocity  of  the  surface  of  the  shaft  is  equal  to  its  cir- 
cumference in  feet  multiplied  by  the  nimiber  of  revolutions 
in  a  second.     And,  if  this  number  of  revolutions  be  n, 

then  V  = 9  ^'^d  the  equation  between  the  stress  and 

^^ 

^     3-78  H 
strain,  reduces  to  cr  =  — - — • 

N 


•v.] 


ON    THE    SHAFTS    Ol-    MILLS, 


therefore  appearB,  that  when  ( )   =  (  o^-j )    t"^ 

in  &OTn  torsion  will  be  equal  to  that  from  lateral  pres- 

.,  147'2       /H  J ,     ,        ,  ,      ,, 

whence  if      .)-  x  ( —  1  be  less  than  w,  the  diameter 

the  shaft  must  he  determined  by  the  rule  for  lateral 
,  if  it  be  greater  than  w,  calculate  the  diameter  by 
rule  for  torsion. 

'or  the  advantage  of  those  who  are  not  much  versed  in 
inlation,  it  may  be  of  use  to  remark,  that  in  common 
t  when  the  stress  in  lbs,  multiplied  by  the  square  of 
length  in  feet,  is  less  than  3,000 ;  the  twisting  strain 
I  be  the  greater;  and  the  reverse. 
1  a  series  of  lyingshafts,  as  in  Fig.  1,  Plate  V.  or  Fig. 
Plate  VIII.  it  is  necessary  to  make  the  journals  equal 
the  twisting  strain  with  the  addition  of  tlie  necessary 
iwance  for  wear ;  hence  it  will  often  be  necessary  to 
1st  the  shafts  to  both  strains  :  but  in  all  instances  where 

lateral  stress  is  less  than  — „     =*(-)'  ^^^  bodies  of  the 
i  need  not  be  greater  than  the  journals. 
fa  other  cases,  as  in  Fig.  1'2,  Plate  III.  the  principal 

Iting  strain  will  be  in  the  part  of  the  shaft  between  the 

(da  A  and  b,  while  the  twisting  strain  on  the  gudgeons 

I  be  equal  only  to  the  friction. 

Having  thus  stated  these  laws,  as  far  as  seems  to  be 

Ksgar}'  for  the  purpose  of  our  present  inquiry,  it  may  be 
Dper  next  to  endeavour  to  apply  them  more  particularly 
pactiee,  with  regard  to  the  proportion  of  shafts. 


SECTION  IV. 


257.  It  was  already  observed,  (Art.  211.)  that  the  gud^ 
or  journals  having  to  support  the  whole  stress  of  the 


236  ON    THE    SHAFTS   OF   MILLS.  [CHAP.V. 

shafts  to  which  they  belong,  their  diameters  being  deter- 
mined, serve  to  guide  in  determining  the  proportions  of  the 
whole  shaft.     They  are  subject  to  wear,  which  the  body 
seldom  is.      They  ought,  therefore,  to  be  sufficiently  large 
to  allow  for  that  wear.     It  frequently  happens,  that  a  shaft 
has  no  lateral  pressure  excepting  that  which  arises  from 
its  own  weight,  for  instance  in  a  line  of  coupled  horizontal 
shafts  conveying  motion  to  a  distance.     In  the  case  of 
vertical  shafts  also  there  is  often  little  or  no  lateral  stress. 
In  such  cases,  when  solid  cast  iron  shafts  are  of  moderate 
lengths,  it  is  found  from  experience,  that  making  them 
square  of  the  same  size  throughout  *,  between  the  joumalsi 
and  the  measuring  a  little  more  on  the  side  than  their 
diameters,  gives  them  sufficient  stiffiiess.   Even  where  there 
is  considerable  lateral  stress  when  the  shafts  are  but  short, 
making  them  in  this  manner  is  found  to  give  sufficient 
stifihess.     This  square  form,  in  many  cases,  affords  great 
convenience  for  hanging  or  fixing  wheels,  pulleys,  &C., 
upon  them. 

258.  The  gudgeons  of  water-wheels  are  often  so  near 
the  wheel,  that  the  stress  is,  in  a  great  measure,  taken  off 
the  shaft.  Hence  some  water-wheels  are  made  without 
shafts,  the  gudgeons  being  fixed  to  the  arms  at  each  side 
of  the  wheel.  The  sole  of  the  wheel,  in  these  cases,  may 
be  considered  as  a  large  hollow  axlef, 

*  See  note  to  Art.  193. 

t  For  a  wheel  of  considerable  breadth  it  will  in  many  instances  be  an  ad- 
vantage to  employ  a  comparatively  small  axis,  and  to  render  the  wheel  firm 
by  arms  and  braces.  But  if  the  axis  be  dispensed  with,  the  same  degree  of 
firmness  will  be  gained  only  by  a  greater  quantity  of  matter. 

The  author  seems  to  have  had  in  view  the  ingenious  method  of  constract- 
ing  a  water-wheel  executed  by  Mr.  Bums,  at  Cartside;  (Dr.  BrenW* 
edition  of  Ferguson's  Lectures,  Vol.  II.  p.  55  ;)  but  the  addition  of  an  ftxw 
would  be  of  great  use  if  it  were  only  to  prevent  the  racking  strain  of  the 
gudgeons  upon  the  cross  arms. 


iP.  v.] 


ON   THE    SHAFTS    OF    .VHLLS. 


237 


59-  1st.  Let  us  now  suppose  a  square  cast  iron  shaft  re- 
to  be  made  eight  feet  long',  with  gudgeons  four 
les  diameter,  having  considerable  lateral  stress  iu  the 
die — were  the  shaft  of  equal  size  throughout,  it  would 
lently  be  weakest  in  the  middle.     Its  section,  in  that 

I,  ought  therefore  to  be  considered.  We  have  already 
1  (Art.  257)  that  square  shafts,  when  short,  are  suffl- 
tly  stiff,  if  made  as  large  as  the  gudgeons ;    this  shaft 

therefore  be  four  inches  square  at  the  end.  Supposing 
section  in  the  middle  enlarged  to  five  inches,  swelling 
I  a  regular  curve  (Art.  187)  from  each  end  (by  Prop. 

then 

The  cube  of  4  is  fri ; 
The  cube  of  5  is  1'25 ; 

is,  nearly  double. — Therefore,  the  strength  of  shafts 

;  inversely  as  their  length,  (Prop.  III.,)  the  section,  as 
R  increased,  would  make  the  eight  feet  shaft  nearly  as 
ng  as  a  shaft  subject  to  the  same  stress,  only  four  feet 
[  and  four  inches  in  the  middle. 

iCt  us  next  see  what  would  be  the  stiffness  of  this  shaft, 
thus  swelled  in  the  middle  to  five  inches.  Then  (hy 
fposition  II.)  the  cube  of  4  is  64,  which,  multiplied  by 
1  equal  to  ^G.  The  cube  of  5  is  125,  multiplied  hy  5, 
equal  to  625  ;  therefore,  the  stiffness  being  inversely  as 
cube  of  the  lengths — 

The  cube  root  of  256  is  6-3. 
The  cube  root  of  625  is  8*5. 

refore,  the  section  as  thus  increased,  would  make  the 
It  foet  shaft  about  as  stiff  as  one  of  six  feet,  subject  to 
same  stress,  and  four  inches  throughout. 
Bo.  Sod.  Let  us  next  suppose  the  point  of  greatest  lateral 
sore  to  be  two  feet  from  one  end. 


238  ON   THE   SHAFTS   OP   MILLS.  [CHAP.  Y. 

From  the  properties  of  the  leyer,  the  gudgeon  next  the 
point  of  greatest  pressure  has  three-fourths  of  the  whole  to 
sustain. 

Now  the  cube  of  the  diameter  of  the  gudgeon  is  64;  miil« 
tiplied  by  2,  is  equal  to  128,  which  represents  the  strength 
of  both  journals  taken  together. 

3.4ths  of  128  is  equal  to  96. 

The  cube  root  of  96  is  4*578,  the  diameter  which  the 
largest  gudgeon  ought  to  have. 
l-4th  of  128  is  32. 

The  cube  root  of  32  is  equal  to  3*174s  the  diametor 
which  the  smallest  gudgeon  ought  to  have,  or  that  farthest 
from  the  point  of  pressure. 

Haying  thus  obtained  the  diameters  of  the  two  gudgeons, 
each  end  of  the  shaft  may  be  made  square,  equal  to  its 
respective  gudgeon.  (Art*  257.) 

The  stress  on  the  shaft  being  as  the  rectangle  of  die 
parts,  (Prop.  V.);  in  this  case  it  is  less  than  the  former,  as 
twelve  is  to  sixteen — ^for  four  feet  multiplied  by  four  fieet  is 
equal  to  16,  which  expresses  the  stress  when  in  the  middle 
— 6  feet  multiplied  by  two  feet  is  equal  to  12,  which  ex- 
presses the  stress  when  two  feet  from  one  end. 

This  shaft  would  therefore  be  as  strong  as  the  fonner, 
if  its  greatest  section  were  reduced  to  4*54. 

For  the  cube  of  5  is  125,  as  16  :  12  ::  125  :  9375,  the 
cube  of  which  is  4'54. 

It  would  be  as  stiff  as  the  former,  if  its  greatest  section 
were  reduced  to  about  4*65  : 

For  the  cube  of  5  is  equal  to  125;  multiplied  by  5  is 
equal  to  625,  which  expresses  the  stiffiiess  of  the  former 
shaft,  as  16  :  12  ::  625  :  468*75,  which  expresses  the 
required  stiffiiess;  and  the  cube  of  4*65  is  equal  to 
100*5444625,  which,  multiplied  by  4*654,  is  equal  to 
467*934,  which  is  nearly  equal  to  the  required  sti&ess. 


t.v.] 


ON    THE    SHAFTS    OF    MILLS. 


•iSii 


examining  shafts  in  this  way,  the  millwright,  aecord- 

;to  the  nature  of  the  case,  will  be  enahled  to  judge  how 

they  should  swell  at  the  place  oi  greatest  sfi-ess. 

.  As  far  as  regards  the  strength  these  examinations 

apply ;  but  not  to  stiffness ;  because  the  rule  for  the 

itifiness  supposes  the  shaft  to  be  everywhere  of  the  same 

section.     And  it  may  be  here  remarked  that  a  cylinder  is 

iiiffer  than  any  figure  which  can  be  inscribed  within  it ; 

hence  there  is  not  that  advantage  in  diminishing  a  shaft 

towards  the  points    of  support  which  many  people    have 

led. 

ipposing  the  reader  to  have  considered  the  principles 

estimating    the   pressure  on  a  shaft  (Art.  ^7)  with 

sufficient  attention  to  be  able  to  ascertain  the  greatest  stress 

upon  one  with  as  much  precision  as  is  necessary  in  these 

inlculations ;  we  shall  here  give  rules  for  computing  the 

diameters  of  different  forms  of  cast  iron  shafts  to  resist 

leral  stress. 


CVLINDBICAL  SHAPT8  OP   CAST   IRON. 


fthe  stress  be  in  the  middle,  and  equal  to  w  cwts.  the 
ire  in  the  middle  not  to  exceed  as  many  hundredths  of 
nch  as  the  shaft  is  feet  in  length  j  then  by  (Art.  255) 
')i  X  H  =  d  the  diameter  in  inches.  That  is,  the  fourth 
of  half  the  stress  in  cwts.  multiplied  by  the  square 
of  the  length  in  feet  is  equal  to  the  diameter  in 
les. 

62.  If  a  cylindrical  shaft  has  no  other  lateral  stress  to 
lin  than  its  own  weight,  then  by  making  the  proper 
titutions,  the  rule  becomes  V'OOy/*  =  d,  the  diameter 
iches. 

iliat  is,  multiply  the  cube  of  the  length  in  feet  by  '007  ; 
the  square  root  of  this  product  is  the  diameter  in 

PS. 

r2 


240 


ON   THE   SHAFTS   OF   MILLS.  [^CHAP.  ¥ 


The  rule  now  stated  may  be  easily  applied  to  any  case 
with  the  advantage  of  both  accuracy  and  simplidtyi  be- 
cause it  enables  us  in  every  instance  to  include  the  effect 
of  the  weight  of  the  shaft  itself ;  when  it  is  computed  in  & 
tabular  form.  Let  the  stress,  supposed  to  be  at  the  mid- 
dle, be  fi  times  the  weight  of  the  shaft,  then  ^Ol^Pn^i 
in  inches. 

263.   Table  of  Shc^  of  Cast  Iron  to  resist  kUerai 
Pressure. 


Length 
in  feet 

Diameter 
in  inches. 

Diameter 
in  inches. 

Diameter 
in  inches. 

Diameter 
in  inches. 

Diameter 
in  inches. 

2 

•237 

•31 

•44 

•54 

•62 

4 

•67 

•88 

124 

1-5 

1-76 

6 

1-23 

1-61 

228 

2^79 

322 

8 

1-9 

2-48 

3^51 

430 

4-96 

10 

2-65 

3^47 

4-9 

6-0 

6-93 

12 

3-48 

4-55 

6-44 

7-89 

910 

14 

4^38 

3-74 

8-12 

9-94 

1V4S 

16 

5-36 

7-01 

9-92 

1215 

1402 

Own  weight 
only. 

Stress  equal 

to  its  own 

weight,  or 

11=1. 

Stress  dou- 
ble its  own 
weight,  or 
ii«2. 

Stress  three 
times  its 

own  weight, 
or  n=8. 

Stress  four 

times  its 

own  weight, 

or  fiB4. 

HOLLOW    CYLINDRICAL   SHAFTS   OP   CAST   IBON. 


264.  To  compute  the  diameter  of  a  shaft  when  it  is  to 
be  a  hollow  cylinder,  it  is  necessary  to  assume  the  ratio 
between  the  diameter  of  the  hollow  part  and  that  of  the 
exterior  of  the  shaft,  in  order  to  avoid  a  complicated  rule. 
Let  D  be  the  exterior  diameter,  and  nd  the  interior  one, 

then  w  bfeinff  in  cwts.  the  rule  will  be  (— ^ -)  =Dthe 

^  ^2(1 -N*/ 

diameter  in  inches.     But  if  the  shaft  supports  n  times  its 

own  weight,  then  V 1-  =  d,  when  the  necessary  sub- 


CHAP,  v.]  OV   THE    SHAFTS    OF   MILLS.  241 

stitutions  are  made.  If  however  some  eonvenient  number 
be  fixed  upon  for  n,  the  rule  may  be  still  further  simplified. 
And  I  thick  one  that  will  be  well  adapted  for  practice  is 
to  make  the  hollow  part  six-tenths  of  the  exterior  dia- 
meter. According  to  this  proportion,  the  rule  is  V  "009 1'n 
=  D.  That  is,  the  cube  of  the  length  in  feet,  multiplied 
oy  "009,  and  also  by  the  number  of  times  the  weight  of 
the  shaft  is  contained  in  the  stress,  then  the  square  root  of 
this  product  is  the  diameter  in  inches. 

The  weight  of  the  shaft  in  lbs.  will  be  very  nearly  equal 
to  the  square  of  the  exterior  diameter  in  inchee,  multiplied 
by  l'(j  times  the  length  in  feet. 

^5.  Table  of  Hollow  Shajis  of  Cast  Iron  to  resist 
■al  Stress. 


diamelor 


Exterior  I  Interii 
inches,  in  inchei. 


12-8 
15-9 
19-4 


\e  we^it  of 
the  abfi. 


[t  will  be  necessary  to  refer  to  the  tables  of  resistance  to 
sion,  previous  to  hxing  on  the  diameter  of  a  shaft. 


inON    SUAPTS    TO    BESISI    LITBRIL    STBEM. 

B66.  If  the  diameter  of  the  shaft  be  calculated  for  cast 
],  and  this  diameter  be  multiplied  by  "935,  the  product  will 
the  diameter  of  a  wrought  iron  shaft  of  equal  stifihess. 


242  ON   THE    SHAFTS   OF   MILLS.  [CHAP.  Y. 

For  in  order  that  shafts  of  these  metals  may  be  equally 
stiff,  the  stiflhess  of  wrought  being  1*3,  when  that  of  cast 
iron  is  unity,  (see  Art.  l62,  note,)  c*  x  1  =  t(;*  x  I'Sor 
c  X  '9S5  =  w ;  where  c  is  the  diameter  of  the  cast  iron 
shaft,  and  w  that  of  the  wrought  iron  one. 


WOODEN   SHAFTS. 


S67.  Suppose  a  cast  iron  shaft  five  inches  square  at  the 
point  of  greatest  lateral  stress — Required  the  size  it  should 
be  when  made  of  oak  to  have  the  same  strength  ? 
The  cube  of  5  is  125. 

Cast  iron  is  four  times  stronger  than  oak,  (see  Banks  on 
Powers  of  Machines,  &c.,  p.  94,)  therefore  125  x  4  = 
500. 

The  cube  root  of  500  is  7*93  inches. 

268.  Shafts  should  be  compared  by  their  stiffiiess,  or 
their  resistance  to  torsion;  but  here  the  question  is  the 
comparative  stifl&iess.     Now  the  stifihess  of  good  oak  is 

to  that  of  cast  iron  as :  1  (Essay  on  Cast  Iron,  Art. 

11-2        ^        ^ 

Oak.)     Therefore  c*xl=o*x  when  the  shafts 

"^  11-2 

are  of  equal  stiffness ;  hence  1*83  c  =  0  ;  where  c  is  the 

diameter  of  the  cast  iron  shaft,  and  0  that  of  the  oak  one. 

And  consequently  an  oak  shaft  should  be  1'8S  times  the 

diameter  of  a  cast  iron  one  to  be  equally  stiff. 

And  the  stiffhess  of  square  shafts  being  as  the  fourth 

powers  of  the  sides  of  the  shafts,  the  side  of  an  oak  shaft 

should  be  1*83  times  that  of  an  iron  one  to  be  of  equal 

stiffness. 

269.  Required  the  size  of  a  fir  shaft  to  have  the  same 
strength  as  one  of  cast  iron  ? 

The  cube  of  5  is  125. 
Cast  iron — 5'5  times  stronger  than  fir,  (Banks,)  there- 


i^ 


r.3  ON    THE    SHAFTS    OF    MILLS.  243 

135    X  5'5  =  687*5  i  the  cube  root  of  687-5  is  equal 

to  8-82 Am. 

270-  The  Btiffhess  of  red  or  yellow  fir  is  to  that  of  cast 

B  —  :  1  (Essay  on  Cast  Iron,  Art,  Fir.)     Therefore 

c  being  the  diameter  of  a  cast  iron  shaft,  and  J"  that  of  a 

fir  one  to  resist  the  same  stress,  c*  x   1  =  /*  x  — -,  or  c  x 

1    =^  that  is  l*7l6  c  =J'.    Whence  it  appears  that  a 

shaft  of  yellow  fir  should  be  1-716  times  the  diameter  of  a 
cast  iron  one,  to  resist  the  same  stress. 

Shafts  that  are  square  should  be  in  the  same  ratio. 

SJl.  In  the  same  manner  we  might   examine  various 

^kinds    of  cylindrical   shafts,   but  what  has  been  already 

^Bud  will  suffice.    In  order  to  shew,  however,  that  considcr- 

^HUe  allowance  should  be  made  for  accidental  stress,  we 

may  mention  the  following  fact  which  lately  occurred. 

272.  The  hollow  shaft  of  a  water-wheel,  in  cousequence 
^^d  one  of  the  gudgeons  getting  loose,  broke  quite  through 
^^kar  one  end,  although  it  was  I6  inches  diameter,  and  two 
^Bches  thick  in  the  shell.  (See  Art.  2l6,  1st  table  of  gud- 
^^Bons.)  The  gudgeons  of  this  shaft  were  not  secured  by 
^^■nches. 

273.  Hollow  cylindrical  shafts  are  often  made  of  equal 
size  throughout,  in  order  to  get  large  flanches,  the  better 
to  secure  the  gudgeon. 

(374.  We  saw  (Art.  254)  that  the  length  of  a  cast  iron 
aH  has  no  influence  on  its  resistance  to  torsion,  and  we 
have  already  said  all  that  seems  to  be  necessary  respecting 
them.  The  case  is,  however,  different  with  regard  to 
wooden  shafts  J  but  we  arc  yet  in  wani  of  facts  to  reduce 


BSIFTS  SUBJECT   TO  TOBSION. 


244  ox   THE    SHAFTS    OF    MILLS.  [CHtf.T. 

the  influence  wliich  their  length  may  have^  to  calciditnL 
As  it  may  give  some  idea  of  this  influence,  we  may  stile 
one  fact  which  came  under  Buchanan's  ohserratioD. 

A  shaft  about  15  feet  long,  made  of  fir,  had  cast  ira 
journals  2iV  diameter ;  one  of  the  journals  broke  firom  fur 
stress,  after  working  about  16  years ;  for  a  connderabb 
time  before  it  broke,  the  resistance  was  equal  to  7  haati 
power,  making  11^  revolutions  per  minute.  It  gradulfy 
wore  until  it  broke ;  when  that  happened,  the  shaft  seeinel 
strained  to  the  utmost,  so  that  it  might  be  reckoned  m 
just  equal  in  strength  to  the  gudgeon ;  the  shaft  was  9} 
inches  square. 

The  cube  of  9|  is  926, 

926  divided  by  55  (Art  269,)  is  16-8 ; 

Cube  root  of  16*8  is  2*6.  That  is,  the  fir  shaft  wodd 
be  equal  to  resist  the  same  lateral  stress  as  a  square  ctst 
iron  shaft  2*6  on  the  side. 

The  cross-tailed  gudgeons  of  wooden  shafts  commoolj 
require  them  to  be  made  sufficiently  large  to  withstand  the 
stress  which  is  brought  upon  them ;  often  indeed,  formerly, 
they  were  much  weakened  by  mortices  cut  through  them 
for  inserting  the  arms  of  wheels ;  this  practice  is  now, 
however,  almost  entirely  abandoned. 

275.  The  power  of  a  shaft  to  resist  torsion,  has  been  cal- 
culated in  comparing  the  resistance  to  torsion  with  that  to 
lateral  stress,  in  Art.  255,  therefore  it  only  remains  to  ap- 
ply the  equation. 


OP    CYLINDRICAL    SHAFTS   OP   CAST    IRON    TO    RESIST    TORSION. 

We  have  found  the  equation  expressing  the  relation  be- 
tween  the  stress  and  strain  to  be  — '- =  d\    the    diar 

N 

meter  in  inches.     But  it  will  be  accurate  enough  for  our 
purpose  to  increase  the  constant  multiplier  to  4,  in  order 


•v.] 


ON    THE    SHAFTS    OF   MILLS. 


24s5 


aider  the  computation  easier ;  with  this  change,  when 
■volutions  per  minute,  we  have  as  a  practical  rule 

=  d',  or(^^^")'  =  d.      That   is,    divide   240 

^       N        ' 

he  number  of  horses'  power  by  the  number  of  revo- 
fotions  per  minute,  and  the  cube  root  of  the  quotient  will 
be  the  diameter  of  the  shaft  in  inches. 

But  the  reader  must  remember  that  when,  in  this  or  any 

her  case,  he  represents  the  power  of  the  first  mover  by  a 

'    rtain  number  of  horses,  he  must  be  certain  to  make  an 

ample  allowance  for  any  temporary  increase  of  the  action 

of  the  moving  power. 

The  following  table  is  calculated  by  the  above  rule,  and 
it  will  be  found  useful  to  compare  with  the  table,  Art. 
'JtiO. 

1^76.  Table  of  Ciflindncal  Shafts  of  Cast  Iron  to  resist 
Torsion. 


Diuneto 

BEVOLOTtO 

Nfl   or  THE 

SHAITB   w 

A   «,NUT1. 

<.rtl>.lbja 
iMbc 

Srw. 

10re». 

SOrar. 

SDrer. 

«re,. 

Wrer. 

■ 

Hor«.- 

H,^, 

Hon«- 

Bona- 

Honci- 

Honu- 

L 

pcncr. 

pOBir. 

1  ' 

0-17 

0-33 

0-66 

0-99 

1-33 

166 

1  ' 

056 

113 

2-25 

3-37 

4-5 

5-62 

1  i 

^■33 

266 

5-33 

7-99 

1066 

13-33 

I  ^ 

2-6 

5-2 

10-4 

156 

20-8 

26-0 

1  ' 

i-5 

9-0 

1600 

27-0 

3G0 

45-0 

I  '' 

7-15 

14-3 

28-G 

+2'9 

37-2 

71-5 

1  ■ 

10-66 

21-33 

42-GG 

640 

85-0 

106-6 

■  10 

20-63 

41-66 

83-33 

125-0 

166-0 

20S-3 

■12 

3600 

72-00 

144-0 

21G-0 

288-0 

360-0 

■n 

63-83 

127-66 

253-33 

383-0 

510-0 

638-3 

|.» 

85-33 

170-66 

341-33 

5120 

C82-0 

853-3 

7.  The  same  table  will  serve  for  hollow  cylindrical 
%  to  resist  torsion,  if  the  diameter  be  multiplied  by 
,  and  the  diameter  of  the  hollow  part  be  six-tenths  of 


^iS  ON   THE    SHAFTS   OF   MILLS.  [cfiAF.  V. 

the  exterior  diameter ;  for  in  that  case  ( -)     =  1'05. 

(See  Art.  254,  note.) 

Indits.  Indiei.     hidiM.     Lidies.   Ii»dic& 
Shafts  of  solid  cylinders ;  diameters       8-         10-         12*         14*         16* 

HoDow  shafts  of  1  exterior  diameter       8*4      10*5       12*6       14*7      16*8 

equal  strength; /interior  diameter       5*  6*3        7*5         6*8      10* 

278.  This  table  applies  to  vertical  shafts,  but  in  hori- 
zontal ones  there  is  an  additional  stress  if  it  be  cmly  from 
their  own  weight,  and  much  more  should  there  be  wheels 
on  the  shaft.  Where  the  lateral  stress  is  small,  it  may  be 
allowed  for  by  adding  something  to  the  diameter  shown  bjr 
the  table,  or  it  may  be  calculated  by  the  rule  at  the  end  at 
this  article. 

Example. — Suppose  a  vertical  shaft  is  to  make  SO  revo- 
lutions per  minute,  the  power  of  the  first  mover  heing 
equal  to  18  horses.  Look  in  the  column  of  horses'  power 
under  20  revolutions ;  and  opposite  18,  the  diameter  will 
be  found  in  the  first  column  to  be  6  inches,  for  cast  iron. 

279.  If  the  shaft  is  to  be  of  wrought  iron,  then  mul- 
tiply the  diameter  found  by  the  rule  or  the  table,  by  0*963. 
(Art.  225,  note.)  Thus,  in  the  above  example,  6  x  0*963 
=  5*778  inches,  the  diameter  of  a  wrought  iron  shaft  to 
make  20  revolutions  per  minute,  when  the  power  of  the 
first  mover  is  equal  to  18  horses. 

280.  If  the  shaft  be  of  oak,  then  the  power  of  oak  being 

when  that  of  cast  iron  is  1  *,  (Art.  268,)  and  the  re- 

11*2 

sistance  to  torsion  being  as  the  cube  of  the  diameter,  we 
have  (11*2)*  =  2*238;  and  multiply  the  diameter  found 
by  the  rule  or  table  for  cast  iron  shafts  by  2*238,  and  it 
will  be  the  diameter  for  an  oak  shaft.  Thus  in  the  pre- 
ceding example,  6  x  2*238  =  13*428  inches  for  the  dia- 

*  The  relative  stiffness  is  used  instead  of  the  relative  strength,  to  reduce 
the  quantity  of  torsion  in  wooden  shafts. 


and 

fct 


CBAP.  V.J  ON    THE    SHAFTS    OF   MILLS.  247 

meter  of  an  oak  shaft  to  make  20  revolutions  per  minute, 
the  first  mover  being  equal  to  18  horses. 

281.  When  fir  is  to  be  used  for  a  shaft,   its  diameter 

should    be    ( — )    times  that  of  a  cast  iron  one  for  the 

same  purpose;  but  ( — )     =  2'06    nearly  ;     therefore    it 

should  be  2'06  times  the  diameter  of  the  cast  iron  one. 

Example. — Let  the  moving  force  be  equal  to  7  horses, 
and  the  number  of  turns  per  minute  11^,  (see  the  case 

died  in  Art.  274.,)  then  by  the  rule  '-^^|^  =  146-09, 

and  the  cube  root  of  146-09  is  5-267  nearly ;  or  practically 
'3  inches  should  be  the  diameter  of  the  shaft  were  it  of 
iron.  And  2-06  x  5-3  =  10-918  inches,  or  nearly  11 
inches  for  the  diameter  of  a  fir  shaft.  It  seems  that  a 
shaft  of  9^  inches  square,  of  fir,  was  found  equal  to  the 
strain ;  and  one  1 1  inches  diameter  is  at  least  it  stronger. 
We  have  made  these  calculations  directly  from  the  theory 
of  equal  cohesion,  but  it  is  so  well  known  a  fact  that  the 
lateral  cohesion  of  fir  is  vastlv  inferior  to  the  direct  cohe- 
sion, that  in  the  rule  for  fir  shafts  to  resist  torsion,  an  in- 
crease of  diameter  should  be  allowed  by  considering  the 
number  of  horses'  power  about  ^  more  than  it  is  intended 
to  be ;  at  least,  till  experiment  shall  have  given  the  pre- 
cise effect  of  lateral  cohesion  in  decreasing  the  force  of 
shafts  to  resist  torsion. 

282.  If  a  shaft  have  to  sustain  both  lateral  stress  and 
ion,  then  the  sum  of  the  straining  forces  must  be  taken  ; 

and  hence  bv  Art.  261,  and  274,  we  have 1  +__  =  rf'. 

N         2rf 

But  in  this  equation  it  is  difficult  to  calculate  the  value  of 

the  diameter,  as  it  is  what  algebraists  call  an  equation  of 

the  fourth  degree.     This  difficulty  may  however  be  easily 

■voided  by  considering  1  d  to  be  2  only,  for  then  the  error 


L      28 
^^kn-sii 


248  ON   THE   SHAFTS   OF   MILLS.  [CHAP.  V. 

will  always  be  in  excess,  except  when  d  is  less  than  unity ; 
and  it  is  much  better  to  be  in  excess  than  defect  Conse- 
quently we  have  as  a  practical  rule  ( +  — )  =rf,the 

diameter  of  the  shaft  in  inches,  when  of  cast  iron.  Where 
/  is  the  length  in  feet  between  the  bearings,  h  the  number 
of  horses  which  are  equal  the  power  of  the  first  moyer,  n 
the  number  of  revolutions  to  be  made  by  the  shaft  in  a 
minute,  and  w  the  lateral  stress  in  cwts. 

Example. — Suppose  that  a  cylindrical  shaft  of  cast  iron 
is  to  make  34  revolutions  per  minute,  the  power  of  the  first 
mover  being  6qual  to  3  horses,  the  length  of  the  shaft  8 
feet,  and  the  lateral  stress  3  cwts.,  when  reduced  to  the 

middle  point,  then  0^2L^^§Ji^y  ^  (gl-lg   +  96)* 

=  (117-18)*  =  4-893  inches. 

283.  With  regard  to  the  making  of  patterns  of  cast  iron 
shafts,  the  reader  is  referred  to  what  has  been  said  in  the 
first  Essay,  relative  to  the  making  of  patterns  for  cast  iron 
wheels,  which  is,  in  a  great  measure,  applicable  to  those  of 
shafts.  Nor  is  there  any  thing  on  this  subject  to  be  added 
here ;  except  to  remind  the  millwright  that  he  make  the 
allowance  for  contraction  of  metal,  of  one-eighth  of  an  inch 
to  the  foot  in  the  pattern. 

284.  The  following  table  contains  the  dimensions  of 
shafts  subject  to  torsion,  and  to  considerable  lateral  pres- 
sure, as  they  were  executed  by  a  respectable  millwright 
It  will  serve  to  shew  the  sizes  of  the  parts  as  found  in 
practice  sufficiently  strong,  and  may  be  foimd  useful  to 
compare  with  those  which  would  be  produced,  calculating 
on  the  principles  laid  down  in  this  Essay. 

Column  6th,  therefore,  shews  the  diameters  which  these 
journals  ought  to  have,  were  400  used  as  the  multiplier. 
(See  Art  Q33.) 


CHAP,  v.]  ON   THE   SHAFTS   OP   UILLS. 


TABLE    OF    SHAFTS. 


Nuno. 

1 

2 

3 

4 

5 

6 

Hemarics. 

l 

.5 

1 

■s 

! 

1 

i. 

Hi! 

lying  shaft. 

Malleable  iron 
lyixigBl^ft. 

20 
18 
16 
U 
12 
10 
B 
8 
5 
4 
3 
2 
1 
6 
5 
4 
3 
2 
1 

20 
22 
22 
24 
25 
25 
27 
28 
30 
32 
34 
46 
40 
28 
30 
32 
3i 
36 
40 

6 
3| 

sj 

5 

5 

*} 

l 

3 

3 

2 

2 

3 

3i 

2 

2 

■i 

1 

11- 
11- 

106 
10- 

9-6 

9- 

9- 

8-6 

8'6 

8- 

8- 

8- 

8- 

8-6 

8- 

8' 

8- 

8- 

7-6 

7 

3 
3i 

7,368 
6,889 
6,621 
6,153 
5,768 
5,428 
4,904 
4,414 
4,061 
3,484 
1,203 
2,802 
2,154 

Feathered 
shafts. 

Square 
s4ftB. 

APPENDIX. 


COHESIVE   STRENGTH    OF   DIFFERENT    METALS. 

285.  "  We  shall  tak^  for  the  measure  of  cohesion  the  num- 
ber of  pounds  avoirdupois  which  are  just  sufBlcient  to  tear 
asunder  a  rod  or  bundle  of  one  inch  square.  From  this  it 
will  be  easy  to  compute  the  strength  corresponding  to  any 
other  dimension. 


"  Gold  cast 


Silver  cast 


f20,« 
(24,1 


"  1st,  Metals. 

lbs. 

20,000 

,000 

(40,000 

(43,000 

Japan        19,500 

Barbary 22,000 

Hungary 31,000 

Anglesea 34,000 

,  Sweden 37,000 

Ironcast i^^^OO 

(59,000 


Copper  cast  < 


APPEND.] 


OK   THE    SHAFTS    OF    MILLS* 


351 


Iron  bar 


Steel  bar 


Tin  cast 


lbs. 

'  Ordinary       ....  68,000 

Stirian 75,000 

Best  Swedish  and  Russian  84,000 

Horse  nails  ....  71>000 

j  Soft 120,000 

( Razor  temper     .     .     .  150,000 

Malacca 3,100 

Banca 3,600 

Block       3,800 

English  block    .     .     .  5,200 

grain    .    .     .  6,500 

860 


Lead  cast 

Regulus  of  Antimony 1,000 

Zinc 2,600 

Bismuth 2,900 

"  The  only  author  who  has  put  it  in  our  power  to  judge 
of  the  propriety  of  his  experiments  is  Muschenbroek.  He 
has  described  his  method  of  trial  minutely,  and  it  seems 
unexceptionable.  The  woods  were  all  formed  into  slips  fit 
for  his  apparatus,  and  part  of  the  slip  was  cut  away  to  a 
parallelopiped  of  i-th  of  an  inch  square,  and  therefore  ^th 
of  a  square  inch  in  section.     The  absolute  strengths  of  a 


square  inch  were  as  follow. 

n>8. 

lbs. 

Locust  tree 

,     .     20,100 

Pomegranate 

.    9,750 

Jnjeb      .    .    . 

.     .     18,500 

Lemon     .    .    . 

.    .    9,250 

BeeeluOak     . 

.     .     17,300 

Tamarind     .    , 

.    .    8,750 

Orange  .    .    . 

,     .     15,500 

Fir      .    .    .    , 

.    .     8,330 

Alder     .    .    . 

.    .     13,900 

Walnut    .    . 

.    .    8,130 

Ehn  .... 

.     .     13,200 

Pitch  Pine    .    , 

.     .    7,640 

Mulberry    .    . 

.     .     12,500 

Quince     .    .     , 

.    .    6,750 

WiDow   .    .    . 

.     .     12,500 

Cypress    .    .    . 

.    .    6,000 

Ash  ...    . 

.     .     12,000 

Poplar      .    .    . 

.    .    5,500 

■a     .    .    . 

.     .     11,800 

Cedar      .    .    . 

.    .    4^880 

tr    .    .    . 

.     .     10^000 

252  ON   THE   SHAFTS    OF   MILLS.  [aPFEND. 

"  Muschenbroek  has  given  a  very  minute  detail  of 
experiments  on  the  ash  and  the  wahiut,  stating  the  weights 
which  were  required  to  tear  asunder  slips  taken  from  the 
four  sides  of  the  tree,  and  on  each  side  in  a  regular  pro- 
gression from  the  centre  to  the  circumference.  The  num. 
hers  of  this  table  corresponding  to  these  two  timbers  may 
therefore  be  considered  as  the  average  of  more  than  50 
trials  made  of  each,  and  he  says  that  all  the  others  were 
made  with  the  same  care.  We  cannot  therefore  see  any 
reason  for  not  confiding  in  the  results ;  yet  they  are  con- 
siderably  higher  than  those  given  by  some  other  writers. 
Pitot  says,  on  the  authority  of  his  own  experiments, 
and  of  those  of  Parent,  that  60  pounds  will  just  tear 
asunder  a  square  line  of  sound  oak,  and  that  it  will  bear 
50  with  safety.  This  gives  8640  for  the  utmost  strength 
of  a  square  inch,  which  is  much  inferior  to  Muschenbroek's 
valuation. 

"  We  may  add  to  these — 

cwt. 

Ivory 16,280 

Bone 5,250 

Horn 8,750 

Whalebone 7,500 

Tooth  of  sea  calf 4,075 

"  The  reader  will  surely  observe  that  these  numbers 
express  something  more  than  the  utmost  cohesion,  for  the 
weights  are  such  as  will  very  quickly,  that  is,  in  a  minute 
or  two,  tear  the  rods  asunder.  It  may  be  said  in  general 
that  two  thirds  of  these  weights  will  sensibly  impair  the 
strength  after  a  considerable  while ;  and  that  one  half  is 
the  utmost  that  can  remain  suspended  at  them,  without 
risk,  for  ever ;  and  it  is  this  last  allotment  that  the  en- 
gineer should  reckon  upon  in  his  constructions.  There  is 
however  a  considerable  difference  in  this  respect  Woods 
of  a  very  straight  fibre,  such  as  fir,  will  be  less  impaired 


APPEND.]  ON   THE   SHAFTS   OF   MILLS.  053 

by  any  load  which  is  not  sufficient  to  break  them  inune- 
diately. 

**  According  to  Emerson,  the  load  which  may  be  safely 
suspended  to  an  inch  square  is  as  follows : 

lbs. 

Iron 76,400 

Brass 35,600 

Hempen  Rope 19)600 

Ivory 15,700 

Oak,  box,  yew,  plum-tree       ....      7f850 

Elm,  ash,  beech        6,070 

Wakut,  plum 5,360 

Red  fir,  holly,  elder,  plane,  crab      .     .      5,000 

Cherry,  hazle 4,760 

Alder,  asp,  birch,  willows      ....       4,290 

Lead 430 

Freestone 91 

"  He  gives  us  a  practical  rule,  that  a  cylinder  whose 
diameter  is  d  inches,  loaded  to  one  fourth  of  its  absolute 
stfength,  will  carry  as  follows : 

cwt. 

Iron 135 

Good  rope 22 

Oak 14 

Rr 9 

**  The  rank  which  the  different  woods  hold  in  this  list 
of  Emerson's  is  very  different  from  what  we  find  in  Mus- 
chenbroek's.  But  precise  measures  must  not  be  expected 
in  this  matter.  It  is  wonderful  that  in  a  matter  of  such 
unqufistionable  importance  the  public  has  not  enabled  some 
persons  of  judgment  to  make  proper  trials." 

S86.  Mr,  Banks  (Powers  of  Machines,  &c.,  p.  94) 
iakeB  iron  at  an  average  to  be  four  times  as  strong  as  oak, 
and  5|  tunes  as  strong  as  deal  or  fir, 

s 


354  ON   THE   SHAFTS   OF   MILLS.  [aPPSMD. 

287*  ^*  According  to  the  experiments  of  various  authors, 
the  cohesive  strength  of  a  square  inch  of  razor  steel  is 
about  150  thousand  pounds,  of  soft  steel  120,  of  wrouglit 
iron  80,  of  cast  iron  50,  of  good  rope  SO,  of  oak,  beedi, 
and  willow  wood,  in  the  direction  of  their  fibres  12,  of  fir 
8,  and  of  lead  about  three  thousand  pounds ;  the  cohesiTe 
strength  of  a  square  inch  of  brick  300,  and  of  freestone 
200 ;  teak  wood,  the  tectona  grandis,  is  said  to  be  still 
stronger  than  oak. 

*^  The  strength  of  different  materials  in  resisting  com- 
pression, is  liable  to  great  variation.  In  steel  and  in  willow 
wood,  the  cohesive  and  repulsive  strength  appear  to  be 
nearly  equal.  Oak  will  suspend  much  more  than  fir,  but  fir 
will  support  twice  as  much  as  oak,  probably  on  account  of 
the  curvature  of  the  fibres  of  oak.  Freestone  has  been 
foimd  to  support  about  2000  pounds  for  each  square  inch; 
oak,  in  some  practical  cases,  more  than  4000. 

*<  The  strongest  wood  of  each  tree  is  neither  at  the  cen- 
tre nor  at  the  circumference,  but  in  the  middle  betwe^ 
both ;  and  in  Europe  it  is  generally  thicker  and  firmer  on 
the  south-east  side  of  the  tree.  Although  iron  is  much 
stronger  than  wood,  yet  it  is  more  liable  to  accidental  im- 
perfections ;  and  when  it  fails,  it  gives  no  warning  of  its 
approaching  fracture.  The  equable  equality  of  steel  may 
be  ascertained  by  corrosion  in  an  acid,  but  there  is  no  easy 
mode  of  detecting  internal  flaws  in  a  bar  of  iron,  and  we 
can  only  rely  on  the  honesty  of  the  workmen  for  its  sound- 
ness. Wood,  when  it  is  crippled,  complains,  or  emits  a 
sound,  and  after  this,  although  it  is  much  weakened,  it 
may  still  retain  strength  enough  to  be.  of  service.**  ♦ 

288.  The  cohesive  force  of  metals  has  been  examined 
by  several  experimental  inquirers,  besides  those  noticed  in 
the  extracts  made  by  our  author,  and  our  knowledge  of 

*  Young's  Nat.  Phil,  toI.  i.  p.  151. 


APPBND.]  ON   THE    SHAFTS   OF   MILLS. 


255 


this  subject  has  been  recently  extended  very  considerably 
by  the  experiments  of  Telford,  Brown,  and  others.  Tred- 
gold  collected  all  the  most  important  experiments  on  the 
cohesive  force  of  metals,  (PhiL  Mag.  voL  1.  p.  421,)  and 
omitting  those  which  are  given  in  the  preceding  articles, 
the  table  is  here  re-arranged  by  him. 


TABLE   OF   EXPERIMENTS   ON   THE   DIRECT   COHESION   OF 

METALS. 


Discriprtoo  of  metal. 

Force  (in 
lba.)that 
would  tear 
asunder  a 
bar  of  one 
inch  square. 

Experimentalist 

Quoted  from 

I.  8TEBL. 

Oast    steel,    pre- 
Tionslj  tilted. 
Cast  steel 

134,256 

63,065 
133,152 

32,973 
127,632 

113,077 
93,964 

85,797 
93,069 

88,972 

85,900 

82,839 

81,901 
80,833 

Rennie. 

Brown. 
Rennie. 

Brown. 
Rennie. 

Siokingen. 

Telford. 

Buffon. 

MuBchenbrogk. 

Idem. 

Idem. 

Idem. 

Idem. 
Soufflot 

Phil.  Mag.  vol.  liii.  p.  167. 

Barlow's  Essay,  p.  234. 
Phil.  Mag.  7ol.  liii.  p.  167. 

Barlow's  Essay,  p.  234. 
Phil.  Mag.  vol.  liiL  p.  167. 

Ann.  de  Chimie,  xxv.  9. 
Barlow's  Essay,  p.  222. 
CEuvres  de  Gauuiey,  ii.  153, 
Intro,  ad  Phil.  Nat  i.  426. 

Rondelet's  L'Art  de  Bitir,  iv. 
500. 

Blister  steel,  re- 
duced    bj    the 
hammer. 

Blister  steel 

Shear   steel,    re- 
duced   by    the 
hammer. 

n.  IfALLBABLB 
IBON. 

Iron  wire 

X von  ^VITO  ••••»«••• 

Iron  wire  ..•• 

German  bar,  mark 
BR,  highest  re- 
sult 

Swed]shbar,high- 
eatresolt 

German  bar,mark 
L,hiriiest  result 

U^  bar,  highest 
remit 

gwiiahbar 

r 

s2 


256 


ON   THE   SHAFTS   OF   MILLS. 


[appekd. 


TABLB  OONTINUID. 


UMcnplioa  of  mettl. 


Oosement  bar, 
highest  result 

Swedish  bar,  re- 
duced by  the 
hammer. 

Common  round 
iron. 

German  bar,  mark 
L. 

Common  Stafford- 
shire bar. 

Conmion  German 
bar. 

Swedish  bar  •••.< 

Oosement  bar,  the 
same. 

Welsh  bar 

Bar  of  the  best 
quality. 

A  bar  of  Welsh, 
one  of  Swedish*, 
and  one  faggoted 
scrap  Iron,  each 
gave  a  result  of 

Liege  bar 

Staffordshire  bar . 

German  bar,  mark 
BR. 

Bar  (mean  of  33 
experiments). 

Russian  old  sable, 
mark  CON. 

English  bar,  re- 
duced by  the 
hammer. 

Welsh  bar  (3 
experiments). 

Bar  of  good  qua- 
lity. 

Swedish  bar,  (3 
experiments). 


Force  (in 
lbf.)that 
would  tear 
asunder  a 
bar  of  one 
inch  square. 


76,697 
72,064 


71,300 
69,538 
69,440 
69,133 
68,728 


66,752 
66,000 

64,960 


Experimentalift 


62,369 
61,600 
61,361 

61,041 

59,472 

55,872 


55,776 
55,000 
53,244 


Muschenbroik. 
Rennie. 


Telford. 
Moschenbro^k. 

Telford. 
Muschenbro^k. 

Idem. 


Telford. 
Rumford. 

Telford. 


Quoted  fron 


Muschenbroek. 

Telford. 
Muschenbroek. 

Perronet 

Brown. 

Rennie. 


Brown. 

Rumford. 

Brown. 


Intro,  ad  FbiL  Nat  L  426. 
PhiL  Mag.  Tol.  liii.  p.  167. 

Barlow's  Essay,  p.  230. 
Intro,  ad  Phil.  Nat  i.  426. 
Barlow's  Essay,  p.  230. 
Intro,  ad  Phil.  Nat  L  426. 


Barlow's  Essay,  p.  228. 
Phil.  Mag.  z.  51. 

Barlow's  Essay,  p.  229. 


Intro,  ad  Phil.  Nat  i.  42$. 
Barlow's  Essay,  p.  229. 
Intro,  ad.  Phil.  Nat  i.  426. 

(Euvres  de  Gaathey,  iL  154. 

Barlow's  Essay,  p.  233. 

Phil.  Mag.  vol.  liii.  p.  167. 


Barlow's  Essay,  p.  233. 
Phil.  Mag.  vol.  x.  p.  51. 
Barlow's  Essay,  p.  232. 


•  Tbo  Swedish  bar  broke  at  a  iaw. 


APFBND.J  ON   THE   SHAFTS  OF   MILLS. 


257 


TABLB  CONTINUED. 


Immi  ipUcMi  01  meliL 


Bar  rfine  grain). . 

—  (medium 
fineness.) 

—  (coarse  grain- 

m.  CAST  IBON. 

Bar,  spec.  gray. 
7-807. 

Bar,  cast  verti- 
cally. 

Bar,  cast  hori- 
sontallj. 

Bar,  Welsh  pig... 

nr.  OOPPBB. 

Wire , 

Wrought  copper, 

ledoced  hj  the 

hammer. 
Cast,       Barhary, 

spec,  ffray.8*  182. 
CSast,  ^pan,spec. 
y.  8-726. 


Force  (m 
lb8.)that 
would  tear 
asunder  a 
bar  of  one 
inch  Mjuaie. 


49,982 
34,081 

20,460 


Ezperimentaliit 


Quoted  ftoin 


68,295 
19,488 
18,656 
16,264 


Rondelet 
Idem. 

Idem. 


grai 

CSlBt 


61,228 
83,792 


22,570 
20,272 
19,072 


MuschenbroSk. 

Rennie. 

Rennie. 

Brown. 


L'Art  de  BAtir,  iy.  502. 


y.  PLATINUM. 

Platmmn  wire, 
^>ecmc  grayity 
80-847. 

Fbdnmn  wire  ... 

Vl.  8XLVXB. 


Iyer  wire.  • 

east,    spec 
giay.  11-091. 


56,473 


52,987 


38,257 
40,902 


Sickinffen. 
Remue. 


Moschenbroek. 
Idem. 


Rennie. 


Intro,  ad  Phil.  Nat  L  417. 

Phil.  Hag.  yoL  liii.  p.  167. 

Idem. 

Barlow's  Essay,  p.  235. 


Ann.  de  Chimie,  zxy.  9. 
Phil.  Mag.  yol.  liiL  p.  167. 

Intro,  ad  PhiL  Nat  i.  417. 


dOySaa 


Monreau. 


Sickingen. 


Sickingen. 
Muschenbroek. 


Biclnngen. 


Phil.  Mag.  yol.  liii.  p.  167. 


Ann.  de  Chimie,  xxy.  8. 


Idem,  p.  9. 


Ann.  de  Chimie,  xxy.  9. 
Intro,  ad  Phil.  Nat  L  417. 


Ann.  de 


xxy.  9. 


358' 


OM  THE   SHAFTS   OF  MILLft. 


[Al 


TABLB  CONTINUED. 


Description  of  metal 

Force  (in 
lbs.)  that 
would  tear 
asunder  a 
bar  of  one 
inch  square. 

Experimentalist. 

Quoted  finom 

Oold  cast,    spec, 
gray.  19*238. 

VIII.   ZINC. 

Zinc  wire 

20,450 

22,551 

16,600 

2,689 

7,129 
6,650 

5,322 

4,736 
3,679 

3,211 

3,328 

3,146 
2,581 

2,547 
1,824 

885 

3,250 
3,008 

1,060 

Moschenbroek. 

Morveau. 

Tredgold. 

Muschenbrogk. 

Morveau. 
Muschenbroek. 

Idem. 

Rennie. 
Muschenbroek. 

Idem. 

Tredgold. 

Muschenbroek. 
Idem. 

Morveau. 

Reunie. 

Muschenbroek. 

Muschenbroek. 
Idem. 

Muschenbroek. 

Intro,  ad  Phil.  Nat  I 
Ann.  de  Chimin?.  Ixxi* 

Phil.  Mag.  vol.  1.  p.  4 
Intro,  ad  PhiL  Nat  i. 

IZ.   TIN. 

Tin  wire 

Ann*  de  CbiTnie^  Ittti- 

English       block, 
cast. 

Intro,  ad  Phil.  Nat  i. 

7-295. 
Cast 

Phil.  Mag.  vol.  liii.  p. 
Intro,  ad  Phil.  Nat  i. 

Phil.  Mag.  vol.  1.  p.  4 
Intro,  ad  Phil.  Nat.  i. 

Banca    tin,   cast, 

specific    gtavity 

7-2165. 
Malacca  tin,  cast, 

specific    gravity 

6-1256. 

X.    LEAD. 

Milled  sheet,  spec, 
grav.  11-407. 
Wire 

Wire,  spec.  grav. 

11-282. 
Wire 

Ann.  de  Chimie.  Ixxi. 

Cast  lead 

Phil.  Mag.  vol.  liii.  p. 
Intro,  ad  Phil.  Nat  i. 

Intro,  ad  Phil.  Nat  i. 

English, spec. 

grav.  11-479. 

XI.    BISMUTH. 

Bismuth,        cast, 
spec.grav.9-810. 

grav.  9-926. 

XII.   ANTIMONY. 

Antimony,     cast, 
8pec.grav.4*500. 

Intro,  ad  Phil.  Nat.  i. 

APPEND.]  ON   THE   SHAFTS   OF   MILLS.  S59 

As  this  table,  the  most  extensive  of  the  kind,  exhibits 
at  one  view  the  chief  results  of  the  experiments  on  the 
direct  cohesion  of  the  metals  in  English  avoirdupois  pounds 
when  the  area  is  a  superficial  inch,  as  well  as  references  to 
the  works  wherein  those  experiments  are  described,  it  will 
be  useful  to  direct  the  labours  of  future  iaquirers  to  such 
experiments  as  are  best  adapted  to  increase  or  correct  our 
kiiowledge  on  this  subject.  It  was  collected  at  various 
times  for  Buchanan's  information,  and  will  be  equally  use- 
ful to  others.  When  experiments  are  not  reduced  to  a 
common  standard,  they  cannot  be  compared  without  much 
labour :  in  the  original  descriptions  of  these  experiments, 
this  has  not  been  done ;  they  are  described  chiefly  as  they 
were  made,  and  for  further  information,  to  these  descrip- 
tions the  reader  must  be  referred.  If  he  be  interested  in 
these  researches,  the  works  of  Muschenbroek,  Rondelet, 
Barlow,  Tredgold,  and  others,  will  afford  him  much  inform- 
ation. 

289*  From  the  simple  metals  we  naturally  look  to  the 
alloys,  some  of  which  are  of  much  importance.  Here  the 
curious  but  important  fact  that  the  union  of  two  metals  pro- 
duces a  compound  of  greater  tenacity  than  either  of  the 
metals  it  is  formed  of,  will  be  noticed. 


260 


ON   THE   SHAFTS   OF   MILLS. 


[appshd. 


TABLE    OF   EXPERIMENTS   ON   THE   DIRECT    COHESION  OF 

ALLOYS. 


AUojof 


Copper 

Ditto... 
Ditto... 
Ditto,.. 
Ditto... 


Pwte. 
..  10 


8 
6 
4 
2 


Tin 


Puts. 
..     1 


Gun  metal,  hard 

Brass,  fine  yellow 

n,  English..  10 
tto 


T 
D 
D 
D 
D 
D 

T 
D 
D 
D 
D 
D 

T 
D 
D 
D 
D 
D 

T 
D 
D 
D 
D 

T 
D 
D 
D 

T 
D 
D 


tto 
tto 
tto 
tto 


8 
6 
4 
2 
1 


tto 


Banca...  10 

8 

tto 6 

tto 4 


ditto 
ditto 
ditto 
ditto 


Lead 
ditto, 
ditto, 
ditto, 
ditto, 
ditto. 


Antimony .... 

ditto 

ditto 

ditto 


tto 2ditto 


tto 


1 


n,  Banca... 


ditto 


10  Bismuth 

tto 4|ditto 

tto 2  ditto 

tto llditto 

tto ijditto 


tto 


llditto 4 


n,  Banca... 

tto 

tto 

tto 

tto 


n,  English.. 

tto 

tto 

tto 

n,  English., 
tto 


10 
2 
1 
1 
1 

8 
4 
2 
1 

1 
3 


Zinc,  Indian 

ditto 

ditto 

ditto 

ditto 


10 


Zinc,  Goslar 

ditto 

ditto 

ditto 


Antimony.... 
ditto 


tto 4  ditto 


Lead,  Scotch      1 

Ditto 2 

Ditto 10 


Bismuth 

ditto 

ditto 


Force  (in  Ibi.) 

that  wmild  tetr 

asunder  a  bar 

of  one  inch 

square. 


d2,0dd 

36,088 

44,071 

35,739 

1,017 

36,368 

17,968 

6,904 
7,922 
7,997 
10,607 
7,470 
7,074 

11,181 
9,881 
12,632 
13,480 
12,029 
3,184 

12,688 
16,692 
14,017 
12,020 
10,013 
7,875 

12,914 
15,025 
15,844 
16,023 
5,671 

10,607 

10,258 

10,964 

9,024 

1,450 

3,184 

11,343 

7,319 
5,840 
2,826 


flrafity  of 
ttie  alloy. 


8-351 

8-392 
8-707 
8-723 


{ 


7-359 
7-276 
7-228 
7-192 
7105 
7-060 

7-576 
7-613 
8076 
8-146 
8-580 
9-009 

7-288 
7-000 
7-321 
7-100 
7130 


7-000 


..••.■••• 

10-931 
11-090 
10-827 


Mascheobroek, 
Intro,  ad  FhiL  NH 
Idem. 
Idem. 
Idem. 
Idem. 

Renide,  Phil.  Tiids. 

Idem. 

MuschenbroeL 
Idenu 
Idem. 
Idem. 
Idem. 
Idem. 

Muachenbroek. 
Idem. 
Idem. 
Idem. 
Idem. 
Idem. 

Muschenbroek. 
Idem. 
Idem. 
Idem. 
Idem. 
Idem. 

Muschenbroek. 
Idem. 
Idem. 
Idem. 
Idem. 

Muschenbroek. 
Idem. 
Idem. 
Idem. 

Muschenbroek. 
Idem. 
Idem. 

Muschenbroek. 
Idem. 
Idem. 


APFEND.3  ON   THE    SHAFTS   OF   MILLS.  26l 

Brass  is  an  alloy  of  copper  and  zinc,  gun  metal  is  an 
alloy  of  copper  and  tin,  sometimes  in  the  proportion  of  96 
parts  of  copper  to  11  parts  of  tin,  but  perhaps  more 
usually  108  parts  of  copper  to  11  parts  of  tin*.  It  will  be 
seen  by  this  table  that  the  proportion  of  six  of  copper  to  one 
of  tin,  is  the  most  tenacious  compound.  A  proportion  very 
near  to  this  is  used  for  bearings,  bushes,  and  some  pur- 
poses in  machinery ;  but  it  is  too  hard  and  brittle  for  many 
uses.  It  is  worthy  of  remark,  that  copper  and  tin  are  soft 
and  malleable  metals,  but  when  combined,  they  form  a 
tenacious,  brittle,  and  hard  alloy.  Both  the  hardness  and 
brittleness  is  increased  by  augmenting  the  proportion  of 
tin. 

Tables  of  the  cohesive  force  of  woods  of  various  kinds 
may  be  seen  in  Tredgold's  Elementary  Principles  of  Car- 
pentry, Sect.  II. ;  also  in  Muschenbroek's  work  above  quoted, 
or  in  Barlow's  Essay  on  the  Strength  of  Timber. 

*  These  numbers  give  the  nearest  chemical  proportions  to  those  in  use 
among  founders.  For  further  information  on  this  subject,  see  the  Art 
Brass,  Supplement  to  Encj.  Brit. 


ESSAY    III. 


ON  THE 


CONSTRUCTION  AND  DURABILITY 


OF  THB 


LONGITUDINAL  CONNEXIONS  OF  SHAFTS,  DENOMINATia) 

COUPLINGS. 


PREFACE. 

Having  treated  of  Wheels  and  Shafts,  both  of  which  may 
be  considered  as  essential  parts  of  mill- work,  the  next  sub- 
ject in  point  of  order  is  the  means  of  connecting  shafts 
longitudinally  J  denominated  couplings,  which  accordingly 
forms  the  subject  of  the  present  essay. 

In  examining  this  subject,  there  have  been  collected  and 
described  a  number  of  different  methods  which  have  been 
employed  in  the  coupling  of  shafts.  These  methods  are 
arranged  under  two  classes;  practical  observations  are 
made  on  each  coupling.  These  observations  are  the  result 
of  Buchanan's  great  experience,  and  that  of  many  others 
well  acquainted  with  the  subject,  with  whom  he  had  taken 
many  opportunities  of  conversing ;  nor  will  these  observa- 
tions be  altogether  without  benefit,  should  they  only  lead 
practical  men  to  make  others  more  extensive,  judicious, 
and  useful. 

A  number  of  facts  relative  to  the  subject  are  also  stated, 
which  will  not  be  without  advantage.     For  however  useful 


ON    COUPLINGS. 

abstract  reasoning  may  be,  yet  a  theory  excluding  some  ap- 
parently trivial  or  minute  circumstances,  is  often  rendered 
altogether  uncertain  in  its  application  to  practice.  Of  this 
we  have  remarkable  instances  in  calculations  made  not 
many  years  ago  by  some  of  the  most  eminent  philosophers 
then  in  Europe,  relative  to  the  motion  of  water  in  canals 
and  pipes.  The  Academy  of  Sciences  at  Paris  over- 
estimated the  quantity  of  water  to  be  delivered  by  an  aque- 
duct so  far,  that  it  was,  when  executed  with  the  greatest 
care,  found  to  be  deficient  in  the  proportion  of  five  to  nine. 
Desaguliers  made  an  error  of  five  parts  out  of  six ;  and 
the  celebrated  M'Laurin  of  ten  parts  out  of  eleven,  in 
estimating  the  water  to  be  conveyed  for  supplying  the  city 
of  Edinburgh*. 

Smeaton,  who  was  certainly  well  able  to  appreciate 
Bcience,  yet  seemed  to  place  more  value  on  the  writings  of 
practical  men  than  those  merely  of  a  theoretical  nature, 
for  he  says,  '*  I  have  myself  always  found  that  exact  ac- 
counts of  buildings  [_and  of  course,  of  other  such  toorks'} 
«luch  were  in  any  degree  remarkable,  and  actually  exc- 
ited, were  much  more  instructive  to  my  mind  than  si/s- 
tnalical  writing."  t  The  professional  learning  of  engineers 
is  now  better  cultivated,  and  they  are  not  required  to 
gatber  the  chief  part  of  their  instruction  from  the  practical 
experiments  of  predecessors.  No  doubt  different  minds 
require  different  modes  of  instruction ;  nevertheless,  a  sys- 
tematic course  of  study  seems  to  be  vastly  preferable  to  a 
desultory  course,  and  also  to  have  been   preferred  by  all 

fhers  either  of  art  or  science.     But  men  like  Smeaton, 
Professor  Robisoo  mentioiiB  these  ctrcumst&nces  in  tbe  Encj'.  Brit.  Art. 
Bitw.     This  lubject  is  now  better  understood.     See  Phil.  Tmns.     Hy- 
dnoUc  InveatigaUon^  subservient  to  an  intended  Croonian  Lecture  on  the 
I,     Motion  of  the  Blood,  by  Dr.  Young.      Read  before  the  Royal  Society, 
f  5,  1S08. 
t  See  Oeecription  of  Bddystone  Lighthoiue,  p.  7. 


S64  ON   COUPLINGS.  [PBEFACE. 

advanced  in  years,  and  fiill  of  occupation,  have  seldom 
either  inclination  or  leisure  to  follow  a  systematic  course. 
They  seek  for  information  only  when  compelled  by  pro- 
fessional difficulties.  They  rely  upon  force  of  genius  to 
supply  the  wants  of  the  moment,  as  an  Indian  hunter  on 
his  exertion  in  the  chase ;  and  like  him  they  have  no  idea 
of  laying  up  a  stock  to  provide  for  unforeseen  exigencies. 
How  different  would  be  the  powers  of  a  man,  of  equal 
genius,  with  the  advantage  of  a  systematic  course  of  study ; 
where  reasoning  was  joined  with  experiment!  Might  we 
not  then  look  forward  to  a  time  when  theory  and  the  laws 
of  nature  would  be  merely  different  terms  for  the  same 
thing  ?  It  is  not  however  to  be  expected  that  this  perfec- 
tion will  ever  be  attained  while  theory  is  confined  to  matter 
divested  of  its  natural  properties. 

But  while  Buchanan  spoke  in  favour  of  practical  works, 
he  did  not  depreciate  those  of  science.  They  may  be  of 
great  mutual  benefit,  and  while  we  listen  with  reverence 
to  the  voice  of  experience*,  by  sound  reasoning  on  her 
dictates,  we  may  extend  and  apply  them  to  purposes  more 
various  and  useful  than  those  to  which  they  originally 
related. 

"The  man  of  science,'*  says  Dr.  Robison,  "  who 
visits  our  great  manufactures,  is  delighted  with  the  inge- 
nuity which  he  observes  in  every  part,  the  innumerable 
inventions  which  come  even  from  individual  artisans,  and 
the  determined  purpose  of  improvement  and  refinement, 
which  he  sees  in  every  workshop.  Every  cotton-mill  ap- 
pears an  academy  of  mechanical  science ;  and  mechanical 
invention  is  spreading  from  these  fountains  over  the  whole 
kingdom;  but  the  philosopher  is  mortified  to  see  this 
ardent  spirit  so  cramped  by  ignorance  of  principle,  and 

•  "  Experience,  slow  preceptress,  teaching  oft 
The  way  to  gloiy  by  miscarriage  foul." 

COWPBR. 


PBEFACE.]  ON   COUPLINGS.  265 

many  of  these  original  and  brilliant  thoughts  obscured  and 
clogged  with  needless  and  even  hurtful  additions,  and  a 
complication  of  machinery  which  checks  improvement  even 
1^  its  appearance  of  ingenuity.  There  is  nothing  in  which 
this  want  of  scientific  education,  this  ignorance  of  principle, 
is  so  frequently  observed,  as  in  the  injudicious  proportion 
of  the  parts  of  machines  and  other  mechanical  structures ; 
proportions,  and  forms  of  parts,  in  which  the  strength  and 
position  are  nowise  regulated  by  the  strains  to  which  they 
are  exposed,  and  where  repeated  failures  have  been  the 
(mly  lessons." 


CHAPTER  I. 


ON   THE   LONGITUDINAL    CONNEXIONS  OF   SHAFTS, 

DENOMINATED   COUPLINGS. 

S90.  It  is  well  known  to  those  who  are  in  any  degree  ac- 
quainted with  mill-work,  that  it  is  very  frequently  necessary 
to  convey  motion  much  farther  than  would  be  practicable 
by  any  one  shaft ;  it  is  therefore  often  requisite  to  connect 
two  or  more  shafts  together*.  These  connexions  are  de- 
nominated couplings  J  and  may  be  divided  into  two  classes ; 
viz.  1st.  Those  having  two  bearmgs :  2nd.  Those  having 
one  bearing.  CoupUngs  having  two  bearings  are  men- 
tioned  first,  because  they  were  long  in  use  before  those 
having  one  bearing,  and  because  they  are,  generally  speak- 
ing,  more  simple  in  their  construction. 


CLASS  I. 


OP   COUPLINGS   WITH    TWO   BEARINGS. 


291.  By  the  hearings  of  shafts  are  meant  the  parts 
which  support  their  pivots,  arbours,  or  journals.  When 
a  coupling  has  double  bearings,  each  shaft  is  supported  by 
two  bridges,  as  represented  a,  b  and  c,  d,  Plate  V.  Fig.  1. 

*  For,  though  it  be  most  desirable  that  machinery  should  be  concen- 
trated as  much  as  possible,  long  ranges  of  lying  shafts  are  unavoidable  m 
largo  mills,  in  cotton,  linen,  and  woollen  manufactories,  breweries,  &c.,  &c. 
Hence  the  efficient  and  durable  connexion  of  these  shafts  is  an  object  of 
considerable  importance. 


»Hap.  1.] 


ON    COUPLINGS. 


COUPLING  I^FiG.  I  AND  2. 


op  THE  SQUARB  COUPLING. 


292.  This  kind  of  coupling  is  formed  by  making  the 
mda  of  the  shafts  to  be  coupled,  square.  These  squares 
iroject  beyond  the  journals,  and  are  in  the  spaces  b  c  and 
i)  E,  between  the  bridges.  One  of  the  squares  is  made  as 
long  as  what  is  called  the  coiipling-box  f.  The  use  of  the 
»upling-box,  which  is  made  of  iron,  is  to  receive  both  the 
iquares,  so  as  that  when  the  one  shaft  is  moved,  the  box 
ionnects  it  with  the  other  shaft  in  such  a  manner  that  they 
lUst  move  together.  But  when  occasion  requires,  the  box 
iaay  be  slipped  back  upon  the  longest  square,  and  so  give 
Sberty  to  take  out  any  one  of  the  shafts,  independently  of 
the  rest,  however  great  the  number  may  be.  A\Tien  the 
iSiafts  are  engaged,  the  box  is  kept  in  its  place  by  the 
pno. 

The  coupling  at  bc  is  represented  as  engaged,  and  at 
IE  as  disengaged. 
Kg.  2.  represents  this  kind  of  coupling  upon  a  larger 
The  same  letters  refer  to  the  same  parts,  as  in 

(Sg.  1. 

Sometimes,  instead  of  the  section  of  the  couplings  form- 
ig  a  square,  as  Fig.  2,  No.  2,  it  is  made  of  an  oblong  form, 
i  represented  in  Fig.  2,  No,  3 ;  but  this  form  is  more 
tffficult  of  execution  than  the  square.  Also,  instead  of 
laving  the  coupling-box  solid,  it  is  frequently  made  in 
two  pieces.  Fig.  2,  No.  4,  in  which  case  it  embraces  the 
whde  length  of  both  squares,  there  being  no  occasion  then 
for  room  to  slide  back  the  box  in  order  to  disengage  the 
couplings. 

OBSERVATIONS. 

S93-  Were  the  axes*  of  these  shafts  truly  in  one  straight 

*  ^xit. — The  lioe,  real  or  imagjjiuy,  that  puses  through  onj  tlung  on 
rttt  it  may  revolT*. 


S68  ON   COUPLINGS*  [CHAP.L 

line,  and  the  squares  made  and  fitted  to  the  hox  with  per- 
fect accuracy,  the  motion  would  he  perfectly  smooth,  but 
in  large  machinery  this  is  almost  impracticahle,  and  eTCD 
if  practicable  when  new,  would  not  long  continue  to  be 
the  case.     The  brasses  wear  unequally,  or  the  firaming 
sinks  more  in  one  place  than  another.     In  some  part  of 
each  revolution,  therefore,  one  or  other,  or  both  the  shafts, 
will  be  lifted  off  their  bearings.     The  two  adjoining  jour- 
nals then  come  to  act  like  one  twisted  piece  of  iron,  and 
must  obviously  occasion  an  unsteady  motion,  and  much 
friction.     This  imperfection  is   sometimes   called  a  Ufl 
This  kind  of  coupling  has,  for  these  reasons,  been  in  a 
great  degree  abandoned  for  null-work.     But  in  small  nur 
chinery,  such  as  in  coupling  the  rollers  of  those  machines 
for  spinning  cotton  called  miUes,  it  is  still  used ;  because 
in  that  kind  of  machinery  it  can  be  executed  with  a  very 
great  degree  of  accuracy,  and  is  not  so  liable  to  wear  out 
of  truth  as  in  larger  works. 

294.  Of  this  species  of  couplings  with  double  bearings 
are  many  varieties,  made  according  to  the  whim  of  dif- 
ferent workmen ;  such,  for  instance,  as  is  represented  in 
Fig.  3,  which  has  a  small  projection  from  each  angle  of 
the  square,  differing  in  no  other  respect  from  the  common 
square  coupling,  and  liable  to  nearly  the  same  faults. 


COUPLING  II.— Fig.  4. 


OP   THE   BOUND  COUPLINO. 


295.  The  round  coupling  has  the  parts  between  the  bridges 
cylindrical.  The  coupling-box  c  is  made  to  fit  those  parts, 
and  to  slip  backward  when  occasion  requires,  as  was  de- 
scribed of  the  square  coupling.  When  the  shafts  are 
engaged,  two  bolts,  de,  and  fg,  pass  through  the  box  at 
right  angles  to  each  other,  and  one  of  them  through  each 


OBSERVATIONS. 


"f  the  shafts ;    this  being   done,   when  the  one  shaft  is 

Imored,  it  win  evidently  carry  round  the  other  along  with  it. 
6.  The  effects  of  the  round  coupling  are  nearly  the 
aa  the  square  coupling,  but  as  the  parts  may  be  all 
turned,  it  is  more  easily  mad,e  true  at  first;  and  when  the 
bolts  which  prevent  it  from  twisting,  wear,  they  may,  with 
little  trouble,  he  renewed ;  but,  as  the  whole  stress  comes 
on  a  small  surface  at  these  holts,  both  the  bolts  themselves 
and  the  holes  very  soon  wear.  For  this  reason,  after  this 
kind  of  coupling  was  some  time  tried  for  the  rollers  in 
cotton  spinning,  it  was  abandoned,  and  the  square  in  which 
ihe  strain  is  diffused  over  a  greater  surface,  substituted. 


COUPLING  in.- 


H^  S97'  Couplings  which  have  no  coupling-boxes,  are  de- 
nominated clutches  or  glands.  They  may,  without  im- 
propriety, come  under  this  class  of  couplings  having  double 

Fig.  5.  represents  a  coupling  of  this  kind,  it  consists  of 
'«■"  crosses,  a  a  and  bb,  one  fixed  to  each  shaft;  bb  has 
ifs  ends  bent  forward,  and  lays  hold  of  a  a,  and  thus 
(urns  round  the  other  shaft. 


OBSERVATIONS. 


i.  Glands  are  an  excellent  mode  of  coupling  for 
e  bearingf!,  and  have  the  advantage  of  throwing  the 
}  further  from  the  centre  of  motion,  than  in  the  square 


^0  ON    COUPLINGS.  [chap.  I, 

coupling  as  commonly  executed;  but  few  workmen  aie 
able  to  execute  glands  with  accuracy,  and  if  this  be  not 
the  case,  they  make  a  very  disagreeable  movement ;  it  may 
therefore  be  not  improper  to  describe  how  this  may  be 
accomplished. 

299*  When  the  axes  of  two  journals  are  put  as  near  to 
a  straight  line  as  possible,  by  observation  fix)m  the  work- 
man's eye,  it  often  happens,  that  they  may  be  in  a  bad 
situation  for  working,  either  because  the  axes  do  not  reaUy 
correspond,  or,  even  though  they  did  correspond,  yet  the 
arms  of  the^  glands  do  not  take  hold  both  together,  or  per- 
haps from  both  causes.  To  adjust  these  arms,  observe,  in 
the  first  place,  when  the  glands  turn  round,  if  one  of  the 
tails  be  in  continual  contact,  and  the  other  tail  always  pre- 
serve an  equal  distance  from  being  in  contact,  in  that  case, 
the  centres  are  perfectly  opposite,  and  axes  in  one  line,  and 
consequently  right ;  but  if  the  distance  of  the  other  point 
of  the  glands  vary  in  its  distance,  or  becomes  so  irr^ular 
as  to  free  the  other  point  of  the  gland,  then,  in  that  case, 
the  centres  are  wrong,  and  must  be  so  adjusted,  that  one 
point  of  the  gland  bear  equally  all  round,  while  the  other 
preserves  an  equal  distance.  The  next  object  is,  to  adjust 
the  points  of  the  glands  so  that  they  be  both  in  contact 
This  may  be  done  by  chipping  and  filing,  and  in  that  case 
they  will  convey  themselves  all  around  in  contact  with  each 
other. 

COUPLING  IV.— Fig.  e. 

BORING  MILL   CLUTCH. 

First  Construction. 

300.  Fig.  6.  represents  a  boring  mill  clutch ;  a  b  c  is  a 
round  plate  of  cast  iron  firmly  fixed  on  the  shaft  m,  next 
the  moving  power  j  de  is  a  lever  connected  with  the  boring 


CHAP.  I.]  ON   COUPLINGS.  271 

shaft  N,  but  which  is  moveable  in  one  direction  on  a  bolt 
at  F,  so  that  it  may  be  moved  to  lay  hold  of  the  projec- 
ticms  H,H,H,H,  on  the  plate  abc,  which  carries  round  the 
lever  d  e  along  with  it,  and  so  moves  the  boring  shaft  n  ; 
by  pulling  the  lever  backward,  the  boring  shaft  n  may  be 
stopped  at  pleasure.  It  is  represented  in  Fig.  6,  No.  1,  as 
disengaged. 

OBSERVATIONS. 

301.  This  kind  of  coupling  is  applicable  to  such  cases 
only  as  have  the  motion  very  slow.  Pressing  on  one  side 
only  of  the  centre,  although  the  axes  of  the  shafts  should 
not  be  exactly  on  a  line,  there  will  be  no  lift.  For  the 
parts,  in  that  case  in  contact,  slide  upon  one  another. 

SOS.  It  is  found,  however,  to  have  a  great  tendency  to 
force  the  bridges  o,p,  on  end.  In  order  to  lessen  this 
tendency,  the  projections  h,  h,  should  be  made  as  far  from 
the  centre  as  conveniency  will  admit. 


COUPLING  v.— Fig.  7. 


BOBINO  MILL   CLUTCH. 


Second  Construction. 


303.  This  coupling,  like  the  first  construction,  has  a 
lever  d  e,  for  disengaging  and  re-engaging ;  but  instead  of 
being  hung  immediately  from  the  end  of  the  shaft,  turns 
on  a  bolt  at  f  in  a  large  cast  iron  plate  ikl.  The  other 
parts  having  the  same  letters  of  reference  as  Fig.  6,  re- 
semble them,  and  are  for  the  same  use.  There  are  three 
spare  sets  of  ears,  qq,  &c.  (which  support  the  lever  near 
the  point  of  pressure)  cast  on  the  plate  ikl,  to  be  used  in 
CMe  of  those  in  action  breaking. 

t2 


^2  ON   COUPLINGS.  [chap.  I. 

OBSERVATIONS. 

304.  The  manner  in  which  the  lever  d  e  is  hung  in  the 
plate  iKL,  at  a  distance  from  the  centre  n,  and  is  supported 
near  the  point  of  pressure  hy  the  ears  qq,  takes  the  stress 
entirely  off  the  holt  f,  and  indeed  off  the  lever  d£>  except 
near  the  ears  qq. 

305.  This,  therefore,  is  evidently  a  stronger  and  better 
clutch,  and  is  accordingly  used  in  horing  the  larger  cylin- 
ders, whereas,  that  on  the  first  construction  is  generally  used 
for  the  smaller  kinds  of  work. 


COUPLING  VI.— Fio.  8. 

306.  This  coupling  is  constructed  by  having  two 
round  cast  iron  plates,  ab  and  cd,  of  the  same  size  aod 
form.  Each  of  these  plates  has  a  part  f,  cut  out, 
fghi^  No.  2,  and  a  projection  l,  corresponding  to  the  part 
cut  out  The  projection  of  the  one  plate  is  inserted  into  the 
opening  in  the  other,  and  thus  serves  to  engage  the  shafts*. 

OBSERVATIONS. 

SO7.  This  coupling  is  simple  and  durable.  It  is  in  fact 
a  species  of  glands^  (Art.  297-)  It  affords  an  excellent 
mode  of  adjustment  The  pivot  and  hole  for  fixing  the 
plates  ought  to  be  round,  and  the  end  of  the  shaft  turned 
along  with  the  journal.  WTien  the  couplings  are  fitted 
into  each  other,  care  should  be  taken  that  those  parts  of 
the  clutch  which  are  opposite  each  other  will  take  hold 
together. 

•  A  slight  Tmnmtion  of  this  coupHug  has  heen  found  a  Tery  good  one. 
It  consists  in  making  the  cin^u)a^  heads  toothed,  so  as  to  fit  together;  the 
te*^  being  wedge-shaped.  This  mode  of  coupling  is  not  affected  bj  any 
Tmnation  or  settlement  of  the  bearii^s,  and  it  is  Tety  doiaUe. 


ON   COUPLINGS. 

Shafts  coupled  upon  tliis  plan,  when  occasion  reqiiires, 
are  easily  moved  out  of  their  places,  independently  of  each 
other. 

COUPLING  vir. 

308.  Fig.  9.  Plate  VI.  represents  the  coupling  link 
used  by  Messrs.  Boulton  and  Watt  in  their  portable  steam- 
engines.  From  one  of  the  arms  a,  of  the  fly-wheel,  pro- 
jects a  strong  iron  pin  p.  On  the  end  of  the  shaft  r,  which 
is  to  be  coupled  to  that  of  the  fly  wheel  a  b,  there  is  a  crank 
c,  which  has  the  same  length  of  an  arm  that  the  pin  is 
distant  from  the  centre  of  the  fly  wheel.  The  pin  and  the 
crank  are  connected  by  a  link  l,  so  that  when  the  fly  shaft 
moves,  it  carries  round  the  other  shaft  r,  along  with  it. 

OBSERVATIONS. 


309.  This  is  a  very  simple  and  durable  contrivance,  and 
if  the  two  shafts  be  only  parallel  to  each  other,  they  may 
work  with  great  smoothness,  although  their  axes  should  be 
in  different  lines,  for  the  links,  in  that  case,  move  without 
any  twisting.  But  if  the  axes  be  not  parallel,  or  if  they 
be  not  in  one  line,  a  twist  vn\\  take  place  at  the  link, 
which  may  be  injurious. 

The  crank  ought  to  be  considerably  longer  than  those 

I  in  general  use,  to  prevent,  as  much  as  may  be,  the  con- 
tj^Dual  drag  on  one  side  of  each  of  the  journals. 

310.  Fig.  10.  represents  a  coupling,  which  is  sometimes 
used  to  convey  motion  from  the  fly  wheel  shaft  a,  of  a 

_  jteam-engine ;  and  is  so  contrived,  that  in  case  the  fly 
^Hioald  torn  the  wrong  way,  the  milUwork  remains  at  rest. 


COUPLING  VIII. 


27^  O^   C0UPLIN08.  [chip.  I. 

and  so  prevents  accidents.  This  ^ect  is  produced  by 
means  of  a  joint  c,  on  the  arm  b,  resembling  the  joint  of  % 
table,  or  of  a  pocket  foot-rule.  When  the  fly  wheel  turns  tin 
proper  way,  the  arm  d,  upon  the  end  of  the  fly  shaft,  acts 
against  the  face  of  the  arm  b,  on  the  mill  shaft  f  ;  and  as 
the  joint  does  not  yield  in  that  direction,  the  mill  shaft  is 
carried  round  by  the  fly  shaft.  But,  if  from  any  acddent, 
the  fly  turns  the  wrong  way,  the  arm  d  strikes  the  back  of 
the  arm  b,  the  joint  yields,  and  the  mill  remains  at  rest 


OBSERVATIONS. 

311.  This  coupling  may  be  made  sufficiently  durable. 
Its  principles  are  nearly  the  same  as  the  boring  mill  clutch, 
Fig.  6.  In  the  Repertory  of  Arts  and  Manufactures,  Vol 
II.  p.  19)  there  is  a  description  of  an  alteration  on  the  cat- 
tle mill,  to  answer  the  purpose  of  these  couplings,  in 
making  the  mill  go  in  its  proper  direction  only.  Buchanan 
was  led  to  this  contrivance  from  the  accidents  to  which 
carding  machines  were  liable  when  driven  by  horses,  and 
in  the  year  1790  he  erected  several  mills  on  that  plan*. 
This  has  since  been  simplified  and  applied  to  thrashing 
mills. 


GENERAL  OBSERVATIONS. 

312.  All  other  couplings  having  two  bearings,  that  he 
recollected  having  seen,  from  the  reasons  already  given,  are 
attended  with  much  friction.  From  that  fault,  they  have 
in  a  great  measure  been  abandoned,  and  those  with  one 
bearing  substituted. 

♦  The  plate  referred  to  in  this  paper  will  serve  to  give  some  idea  of  the 
manner  in  which  mill-work  was  constructed  ahout  the  year  1790,  in  this 
part  of  the  island.  Cast  iron  was  then  but  little  used  in  cattle  mills.  See 
ItUrodwHon  to  Buo^  Seeandy  on  the  Skaftg  ^  Milk. 


CHAP.  I.]  ON   COUPLINGS.  275 

SIS.  It  is,  however,  proper  here  to  remark,  that  although 
the  friction,  and  expense  of  erection  of  couplmgs  with  one 
bearing,  are  considerably  less  than  those  having  two  bear- 
ings, where  the  chief  object  is  to  convey  motion  to  a  dis- 
tance, or  where  there  is  no  great  weight  or  lateral  stress  on 
the  shafts,  yet  all  circumstances  should  be  duly  considered 
before  adopting  either  plan :  for  if  heavy  drums  or  wheels 
are  to  be  placed  near  both  ends  of  the  shafts,  or  any  other 
thing  that  will  occasion  much  lateral  pressure,  it  will  be 
advisable  to  use  two  bearings.  In  the  next  chapter,  are 
examined  couplings  having  one  bearing. 


CHAPTER  IL 


CLASS  II. 


OF   COUPLINGS   HAVING   ONE   BEARING. 


SECTION  I. 


314.  This  class  of  couplings,  when  properly  constructed, 
has,  to  a  certain  degree,  the  property  of  being  flexible  in 
all  directions,  like  the  well  known  contrivance  invented  by 
Dr.  Hook,  (Fig.  11,)  called  the  universal  joint.    This  joint 
is  sometimes  constructed  by  a  cross,  as  represented  in  the 
figure,  and  sometimes  with  its  four  pivots  fastened  at  right 
angles,  upon  the  circumference  of  a  hoop,  or  on  the  sur- 
face of  a  solid  ball.     The  moving  parts  are  evidently  alike 
in  all  these  cases. 

315.  It  is  sometimes  applied  to  communicate  motion, 
instead  of  bevelled  gear,  when  the  angle  does  not  exceed 
30  or  40  degrees,  and  where  the  number  of  revolutions  is 
to  be  continued  the  same ;  also,  where  equality  of  motion 
is  not  required,  for  as  it  recedes  from  a  right  line,  its  mo- 
tion becomes  irregular.  The  property  of  the  universal 
joint  for  conveying  angular  motion,  is  of  great  use  when  it 
can  be  attained  in  couplings,  in  order  to  allow  for  the  in- 
accuracy which  arises  from  the  settling  of  the  framing  or 
the  wearing  of  the  brasses.  It  need  not,  however,  yield 
further  than  what  is  really  necessary  for  that  purpose, 
which  is  so  little,  that  no  irregularity  of  motion  that  can 
be  hurtful  in  practice  can  arise. 


CHAP.  II. ]  ON    COUPLINGS.  277 

316.  The  disadvantage  of  the  universal  joint,  as  repre- 
sented by  Fig.  10,  and  as  commonly  made,  is,  that  it  has 
not  sufficient  strength  to  resist  great  strains.  There  is, 
however,  afterwards  described  a  modification  of  it,  well 
adapted  to  bear  very  considerable  stress.   (See  Fig.  18,  Art. 

829.) 

8I7.  Most  part  of  the  couplings  described  in  Chap.  I. 
may,  with  some  small  addition  or  alteration,  be  converted  into 
couplings  having  one  bearing.  Of  this  description  is  the 
sqiuire  coupling. 


COUPLING  IX. 

\ 

THE  SQUARE   COUPLING. 

318.  It  consists  of  a  square  coupling  box  c.  Fig.  12, 
which  fits  a  square  on  the  end  of  each  shaft.  The  square 
of  the  shaft  a  is  close  to  its  joumaL  The  square  of  the 
shaft  B  is  at  the  end  furthest  from  its  journal. 

In  order  to  support  the  square  of  b,  and  keep  it  on  a 
line  with  a,  there  is  a  round  hole  d,  bored  out  of  the 
centre  of  the  square  of  b  ;  to  fit  this  hole,  there  is  a  round 
projection  d,  from  the  square  of  a.  Instead  of  having  the 
projection  d,  sometimes  there  is  a  hole  in  the  end  of  both 
shafts,  into  which  is  fitted  an  iron  or  steel  dowel  or  pin. 

The  coupling  box  c,  covers  both  squares.  It  is  kept  in 
its  place  by  two  iron  pins  or  bolts,  g,  g,  one  of  which  passes 
through  each  shaft.  These  bolts  also  serve  to  keep  the 
shafts  from  withdrawing  from  each  other. 


OBSERVATIONS. 


dl9.  This  coupling  is  used  with  good  effect  in  conveying 
motion  through  a  great  length  of  shafts,  where  there  is  but 


278  ON  COUPLINGS.  [chap.il 

little  lateral  pressure;   but  where  there  is  much  ktoal 
pressure,  for  instance  in  drum^htyUi  which  give  motioD  to 
carding  engines  by  belts,  it  has  been  found,  that  the  round 
projection  and  socket  soon  wear  and  get  loose.     This  fknlt 
perhaps  arose  from  the  square  being  too  small  to  admit  the 
projection,  and  the  socket  to  be  sufficiently  large.    But 
though  this  be  a  better  mode  of  coupling  than  many  m 
use,  yet  the  difficulty  of  having  the  squares  accurately  ad- 
justed is  great,  as  is  also  the  fitting  of  the  inside  of  the 
coupling  box,  from  the  ordinary  mode  of  casting.    For 
these  reasons,  in  many  cases  it  is  very  liable  to  lifting  or 
straining. 

320.  In  England,  most  of  the  rollers  of  those  machines, 
denominated  mides*,  for  spinning  cotton,  have  but  one 
bearing ;  the  end  of  one  roller  being  squared  and  inserted 
into  the  end  of  that  next  to  it,  as  represented  in  Fig.  13. 
But  as  there  is  a  great  lateral  pressure,  the  end  of  the 
roller  is  liable  to  wear  and  become  wide,  and  occasion  a 
hobbling  and  inaccurate  motion.  From  the  accuracy  re- 
quired in  miUe  and  throstle  f  rollers,  it  is  not  only  necessary 
that  the  axes  be  exactly  in  one  line,  and  accurate  in  their 
diameters,  but  also  that  they  run  true  without  hobbling  or 
jolting.  Buchanan  had  never  seen  any  method  by  which 
this  was  attainable  in  one  bearing ;  even  though  the  utmost 
attention  of  the  most  accurate  workmen  had  been  bestowed 

*  The  mule  was  invented  about  the  year  1777,  by  Samuel  Crompton, 
fonnerly  of  Hall-in-the-Wood,  near  Bolton,  in  Lancashire,  a  person  of  yeiy 
great  ingenuity,  and  to  whom  the  country  is  indebted  for  many  other  useful 
improvements ;  this  machine  probably  received  its  name  for  having  rollers 
like  Arkwright's  machine,  at  the  same  time  that  it  retained  the  carriage  and 
spindles  of  hand-spinning  machines  called  common  jennies. 

In  1769,  Arkwright  obtained  his  patent  for  spinning,  and  in  1775,  for 
preparing  cotton  by  machinery. 

t  ThrostUy  a  machine  for  spinning  cotton,  compounded  from  the  inven- 
tions of  Arkwright  and  Crompton. 


BAP.  II.] 


ON    COUPLINGS. 


®79 


i  accomplish  this  end.  In  Scotland,  two  bearings  are 
nerally  used  in  the  coupling  of  all  rollers  employed  in 
ning  cotton. 

COUPLING  X. 

OP   THE  BOCND  COUFLlNa. 

3€1.  This  coupling  having  only  one  bearing,  is  repre- 
gented  by  Fig.  13,  and  has  similar  additions  to  those  of  the 
last  described  coupling,  Fig.  12,  and  which  will  appear 
sufficiently  from  the  figures,  without  further  description. 


COUPLING  XI. 


^^  OBSERVATIONS. 

322.  The  observations  which  were  already  made  on  the 
round  coupling  with  two  bearings,  (Art.  296,  Fig.  4,)  are 
in  a  great  measure  applicable  here,  as  are  those  relative  to 
lateral  stress,  in  the  observations  (Art.  319,)  on  Coupling 
IX. 

^(  0X3.  Fig.  14  represents  a  coupling  used  in  several  of 
■  tile  mills  at  Manchester.     It  consists  of  a  scarfed  joint, 
(like  that  used  in  carpentry,)  and  when  the  shafts  are  cn- 
gaged,  it  is  firmly  bolted,  as  shewn  in  the  figure. 

B  324.  It  was  probably  owing  to  the  defect  relative  to  lateral 
stress  mentioned  in  observations,  (Art.  319,)  Coupling  IX., 
that  this  contrivance  was  adopted.  But  it  seems  to  me  to 
have  all  the  defects  attending  the  solid  shaft  witli  more 

■Mhan  tiro  bearings,  namely,  that  it  could  not  for  any  length 


OBSERVATIONS. 


S8Q  ON    COUPLINGS*  [CHAP.U. 

of  time  move  properly  on  all  its  bearings,  without  a  am- 
tinual  bending  of  the  solid  metal,  which  would  occasicm  a 
waste  of  power,  besides  a  great  risk  of  breaking  the  shaft. 


COUPLING  XIL 

325.  The  coupling  represented  by  Fig.  15,  Plate  VIL, 
is  only  a  variety  of  Coupling  XI.,  and,  therefore,  the  same 
observations  are  applicable  to  this  also. 


COUPLING  XIII. 


326.  That  shewn  in  Fig.  16  may  also  be  considered  as 
only  another  modification  of  Coupling  XI.,  the  one  shaft 
being  firmly  fixed  to  the  other  by  flanches  and  bolts.  The 
same  observations  are  also  applicable  here. 


COUPLING  XIV.— Pio.  17. 

327-  Has  the  bearing  and  the  joint  of  the  coupling  at 
the  same  parts  of  the  shaft,  so  that  the  journal  is  not  solid, 
but  composed  of  the  ends  of  each  shaft,  which  are  there 
formed  into  quadrants.  Each  shaft  having  two  projections, 
corresponding  to  two  recesses  in  the  other,  one  of  the  pro- 
jections is  represented  at  a,  No.  1,  as  engaged ;  No.  2  re- 
presents the  coupling  as  disengaged. 


OBSERVATIONS. 

328.  This  is  obviously  so  bad  a  contrivance,  that  little 
need  be  said  with  regard  to  it.  The  bad  effects  of  it 
were  lately  brought  under  Buchanan's  observation  at  an 
extensive  set  of  calico  printing  works.  It  was  there  ap- 
plied to  drive  dash- wheels  for  washing  calicoes,  and  had 
been  attended  with  great  trouble  and  expense;  for  the 


ClUr.  II.]  ON    COUPLINGS,  281 

shafts  ven  frequently  broke  at  the  couplings,  and  were 
thns  rendered  useless.  Besides,  while  they  did  last,  the 
couplings  thus  formed  in  the  journals,  acted  as  a  kind  of 
cutters,  producing  great  friction,  and  tearing  down  the  brass. 


COUPLING  XV— Fig.  18. 

329.  This  consists  of  three  distinct  parts,  a  a,  BB,andcc; 
A  A  and  B  B  are,  each  of  them,  firmly  fixed  to  its  respective 
shaft,  and  are  so  formed,  that  they  join  partly  into  one 
another,  in  some  degree,  like  the  parts  of  a  common  papier 
mnchi  snuff-box, 

c  c  consists  of  a  solid  ring  of  cast  iron,    into  which  is 
screwed  four  strong  stfcl  or  wrought  iron  pins  d,  d,  d,  d. 
[lese  pins  serve  to  act  as  the  four  pivots  of  the  universal 
it  already  described  in  Fig,  11, 

I A  and  BB  are  so  contrived,  that  each  of  them  lays  hold 
two  opposite  iron  pine,  the  whole  combined  thus,  form- 
an  universal  joint.  There  is  a  space  of  about  a  quar- 
of  an  inch  left  between  a  a  and  b  b,  in  order  to  allow 
lie  joints  to  play;  and  a  a  and  B  B  are  commonly  made 
;ii)OUt  It'  inches  diameter  outside. 

There  is  a  small  mortise  in  each  pin,  to  receive  a  cot- 
teril,  to  prevent  the  pin  from  coming  out  in  the  course  of 
working. 

In  the  plate,  so  many  different  views  of  this  coupling 
given,  that  it  ia  hoped  no  further  description  will  be 
]uired, 

OBSERVATIONS. 

■  830.  Tliis  coupling  is  somewhat  expensive  in  its  first 
■cliuo  ;  but  it  being  evidently,  as  far  as  is  necessary  in 
.eh  a  coupling,  a  complete  universal  joint,  from  its  dura- 
JbtT  and  saving  of  power  which  its  pliancy  must  occasion. 


J 


OH  COUFLIN68.  [CHAP.  IL 


it  seems  to  be  perbaps  the  best  thing  ci  the  kind  that  bas 
yet  come  under  Buchanan's  observatioiu 


COUPLING  XVI.— Pio.  19,  Plates  VIL  and  VIII. 

SSI.  This  is  a  kind  of  coupling  which  was  in  Buchan- 
an's time  executed  at  Manchester.  The  coupling  box  c  is 
made  very  long,  and  is  square  in  the  inside,  excepting  at 
D£,  where  there  is  a  kind  of  partition,  with  a  large  round 
hole  truly  bored  in  it.  Into  this  hole,  each  of  the  shafts 
A  and  B,  are  accurately  fitted.  The  round  part  of  the 
shaft  B,  however,  is  made  so  long,  as  to  allow  the  liberty  of 
slipping  back  the  coupling  box,  in  order  to  disengage  the 
shafts.  When  engaged,  the  box  is  kept  in  its  place  bj  the 
pin  B.    H  represents  the  joumaL 


OBSERVATIONS. 

332.  The  principles  of  this  coupling  are  nearly  the 
same  as  those  of  the  common  square  coupling,  (X.  Fig.  l,) 
(Art.  292,)  but  the  greater  length  of  the  box,  as  well  as 
the  greater  strength  of  the  round  parts  intended  to  keep 
the  axis  true,  give  this  last  coupling  very  material  advan- 
tage. 

333.  This  kiud  of  coupling  has  another  advantage  which 
the  greatest  part  of  single  bearing  couplings  have  not, 
viz. :  when  the  coupling-box  is  shifted  off,  any  shaft  may 
be  taken  out,  without  affecting  those  adjoining. 


COUPLING  XVIL 


334.  Fig.  20,  Plate  VIII.  represents  a  coupling  used  in 
one  of  the  cotton  mills  last  erected,  and  one  of  the  most 
extensive  at  Glasgow.  The  external  part,  coupling  box  c, 
is  cylindrical.     On  the  inside,  the  parts  a,  a,  a,  a  project, 


CHAP*  n*]  ON   COUPLINGS.  28S 

and  are  fitted  into  the  shafts,  which  have  similar  projec- 
tions, E,  £,  £,  £  fitted  into  the  coupling  hox.  Through  the 
centre  of  the  coupling  box  there  passes  a  bolt  h  h,  to  keep 
it  in  its  place ;  and,  at  the  further  ends  of  the  row  of  lying 
shafts,  they  are  kept  together  by  working  against  a  kind  of 
step  formed  of  brass.  Were  this  not  the  case,  two  bolts 
would  be  necessary  in  each  coupling,  to  keep  the  shafts 
from  separating. 

OBSERVATIONS. 

335.  The  advantages  of  this  kind  of  coupling,  seem  to 
be  these  two:  Ist.  The  projecting  parts  e,  e,  &c.,  can  be 
accurately  turned  and  fitted  to  the  coupling  box.  2d. 
These  projecting  parts  tending  to  the  centre,  are  strong, 
and  little  liable  to  wear. 


SECTION  11. 

OP   THB  COUPLINGS  OP   UPRIOHT  8HAPT8. 

336.  Hitherto  we  have  considered  couplings  for  lying 
shafts  only.  But  of  upright  shafts,  little  need  be  said; 
having  in  general  little  lateral  pressure,  they  are  seldom 
made  with  two  bearings ;  so  that  by  placing  any  one  of  the 
coupliiigs  with  one  bearing,  already  mentioned,  in  a  verti- 
cal position,  an  idea  will  be  obtained  of  the  mode  of 
coupling  upright  shafts. 

The  square  coupling,  (IX.,  Fig.  12,  Plate  VI.,)  for  in- 
stance, may  easily  be  applied  to  an  upright  shaft. 

COUPLING  XVIII.— Pio.  21. 

397*  A  represents  the  journal  of  the  lower  shaft,  (which 
if  almost  always  that  which  has  the  bearing,)  b  is  the  lower 
mA:  of  the  upper  shaft;,  and  c  the  coupling  box. 


284  ON  COUPLINGS.  I^CHAP.n. 


COUPLING  XIX.— Fig.  22. 

338.  A  represents  part  of  the  under  shaft,  b  the  lower 
end  of  the  upper  shaft,  c  the  journal  The  termination  of 
A  is  made  square,  to  correspond  with  which  there  is  a 
socket  formed  in  b,  which  answers  the  purpose  of  a 
coupling  hox. 


OBSERVATIONS. 


339*  This  coupling  is  often  used  for  light  work,  particu- 
larly in  flour-miUs,  for  connecting  the  feeder  with  the  top 
of  the  stone-spindle. 


COUPLING  XX.— Fig.  23. 


340.  A  represents  the  lower  shaft,  d  the  upper,  which  is 
above  the  joumaL  Projecting  and  receding  quadrants,  the 
same  as  in  Fig.  17,  serve  to  connect  the  shafts. 


OBSERVATIONS. 

341.  This  is  a  very  good  and  simple  mode  of  coupling 
upright  shafts.  By  their  own  weight,  together  with  that  of 
wheels  that  may  be  on  them,  the  projecting  and  receding 
quadrants  d,  d,  are  pressed  home  into  their  holds,  and  are 
not  subject  to  get  loose  in  their  sockets  or  clutches,  which 
would  be  the  case  were  the  shafts  lying  horizontally.  A 
great  many  other  schemes  for  couplings  have  been  intro- 
duced ;  most  of  them  ingenious ;  but  the  late  Mr.  Tred- 
gold  had  not  examined  any,  which  are  sufficiently  simple 
and  likely  to  be  durable,  to  offer  here  as  improvements. 


CHAPTER    in, 


GENERAL  OBSERVATIONS. 


,  It  may  be  proper  to  observe,  that  the  larger  the  parts 
f  the  coupling  can  conveniently  be  made  the  better.     In 
ber  words,  the  further  the  point  of  stress  is  from  the  axis, 
B  couplings  will  be  the  more  durable.     This  being  a 
I  to  which  too  little  attention  in  practice  is  paid,  it 
it  be  improper  here,  in  a  popular  way,  to  endeavour 
lain  the  reasons  of  this  greater  durability. 
The  strain  on  the  point  of  stress  is  inversely  as  the 
city  of  that  point.     (Essay  II.)     Now  the  revolutions 
J  the  same  in  a  given  time,  the  further  the  point  of 
i  is  from  the  axis,  the  greater  will  be  the  velocity  of 
!  point,  and,  consequently,  the  less  the  stress.     Expe- 
mce  has  taught  this  to  those  unacquainted  with  science ; 
'  every  one    knows  that  a  handspike,   capstan-bar,   or 
r  similar  lever,  requires  to  be  largest  near  the  fulcrum, 
i  point  on  which  it  turns  in  raising  a  weight,)  and  may 
e  diminished  in  proportion  to  the  distance  from  the  centre 
of  motion. 

344.  Thus,  for  example,  at  two  feet  from  the  centre,  the 
Bfcresa  is  only  one  half  of  what  it  is  at  one  foot  from  the 
centre.  This  is  almost  evident  to  the  feeling,  from  the 
force  the  hand  has  to  apply  at  those  different  distances. 
Now,  the  larger  the  parts  are,  the  stress  must  be  thrown 
the  farther  from  the  centre  of  motion,  and  the  acting  parts 
will  therefore  be  the  more  durable.  Hence  also  the  ad- 
vantage in  niachincr)'  of  having  targe  wheels  and  large 


II 


286  ON   COUPLINGS.  [chap.  IIL 

pulleys.     (See   Essay  I.,    Oeneral    Observations  on  Ae 
Wheel' Work  of  Mills,  Art.  101.) 

345.  It  is  also  proper  to  observe,  that  when  tbere 
is  a  long  line  of  shafts,  the  couplings,  where  there  is  only 
one  bearing,  should,  if  practicable,  be  so  arranged  as  that 
the  unsupported  end  of  the  shaft  should  be  as  far  as  may 
be  from  the  part  subject  to  lateral  pressure.  For  instance, 
in  Fig.  21,  Plate  VIII.,  the  couplings  and  journals  m 
better  as  there  represented,  than  had  they  been  at  a  a,  in 
the  middle,  between  the  wheels. 

346.  The  oiling  of  couplings  is  found  to  render  them 
more  durable.  This  fact  has  been  fiilly  ascertained  in  one 
of  the  most  extensive  cotton  manufactories  in  Britain,  in 
the  machinery  of  which  the  couplings  were  formerly  very 
liable  to  wear,  but,  since  using  oil,  they  have  been  found 
sufficiently  durable.  The  squares  of  couplings  nearest  the 
journals,  from  accidentally  getting  oil,  are  also  least  worn. 

347*  A  fly-wheel  is  often  of  use  in  a  long  line  of  coupled 
shafts.  In  the  vicinity  of  Glasgow,  motion  was  conveyed 
from  a  steam-engine  by  means  of  lymg  shafts,  to  the  dis- 
tance  of  ninety-three  yards.  When  those  shafts  were  first 
tried,  from  the  elasticity  or  spring  of  so  great*  a  length  of 
shafts,  and  the  play  of  the  couplings,  the  motion  at  the 
further  end  was  so  very  irregular,  that  it  could  not  be  ap- 
plied to  work  a  "  calender."  A  fly-wheel  near  the  "  ca- 
lender," connected  with  the  lying  shaft  by  pulleys  and  a 
belt,  was  resorted  to  in  order  to  cure  this  evil,  and  this 
simple  contrivance  had  the  desired  effect ;  for  the  calender 
ever  after  gave  satisfaction  in  its  work. 

A  table  respecting  the  dimensions,  stress,  and  durability 
of  couplings  is  annexed. 


1 


CHAP.  III.]  ON    COUPLINGS. 


487 


SiS.  Eacts  respecttTig  Couplings. 


1 

s 

3 

4 

5 

e 

II 

1 

s. 

1 

t 

■s 

1 

1 
1 
1 

J 

! 

Is. 

A,  Cast  iran.     Square  coupUog,  one 
bearing,    (see   Coupling   IX.    Fig 
12,)  coupling  next  the  steam-en- 
giiie,  worn  off  each  uigle  of  the 
square  about  three-fourths  of  an 
inch;  the  square  was  originally  5 
inc&eii  the  Imx  10  inches  long. .. 
B,  CmI  iron.     Same  Une   of  shafts 
further  on ;  all  things  but  the  re- 
sistance the  same.      Some  of  the 
omplisge    not    perceptibly    worn 
others  worn  off  about  three-eighths 

IC 

6 
17 

e 

1 

12 
9 

12 

e 

40 

40 

3S 
38 

50 
55 

50 
50 

7 

7 

5 

5 

5 
8 
4 

7 
7 

s 

8 
3 
2 

4 
31 

in 

10 

12 

6 

5 
8 
8 
6 
3 

■40 

■20 

■44 

■1.'. 

-02 
■24 
■16 
-24 
■12 

C,  Cast  iron.   Same  kind  of  couplings 
•e  A  and  B,  sqoaie  originally  6 
bchea,  box  1 2  inches  long,  not  per- 

D,  Cast  iron.    Same  kmd  of  coupling 

E,  Wrou^t  iron.     Same  kind,  much 

W,  Cart  iron.     Same  kind,  not  worn 
&.  Cost  iron.     Worn  one  half  inch.. 
H,  Coat  iron.     Not  perceptibly  worn 
1,  Cast  iron.     Not  perceptibly  worn 

DXtCRIPTION   OF  THE   TABLE. 

Colonm  1  contains  tlie  resistance  in  horses'  power. 
Colunm  S  contains  the  revolutions  per  minute. 
Column  3  contains  the  years  at  work. 
Column^  4  contsuos  the  side  of  the  square  in  inches. 
Column  5  contains  the  length  of  the  box. 

U  2 


288  ON  COUPLINGS.  [cHAP.m. 

Column  6  is  found  by  dividing  the  power  by  the  revo- 
lutions per  minute,  which  represents  the  ami- 
parative  stress,  (see  Essay  II.  Chap.  IV.) 


OBSERVATIONS. 

I. 

349*  These  couplings  do  not  seem  to  have  been  durable 
in  proportion  to  their  stress ;  but  this  may  be  in  part  at 
least  accounted  for,  from  difference  of  workmanship,  and 
of  degrees  of  hardness  of  metal,  or  perhaps  accidentally 
getting  oil. 

II. 

S50.  Two  circumstances  must  materially  affect  the  da- 
r ability  of  couplings.  1.  The  extent  of  surfaces  in  con- 
tact at  the  place  of  pressure.  S.  The  distance  of  the  sur- 
face of  pressure  from  the  centre  of  motion. 

It  is  probable,  therefore,  that,  all  other  circumstances 
being  the  same,  the  durability  of  couplings  increases  in  a 
ratio  compounded  of  those  two  circumstances,  or  nearly  as 
the  squares  of  the  sides  of  such  couplings  as  have  square 
coupling  boxes. 

Thus,  for  example,  b,  in  the  table,  is  5  inches  on  the 
side ;  admitting  the  above  ratio  to  be  near  the  truth,  in 
order  to  have  the  case  a  made  in  proportion  to  b,  it  should 
be  increased  to  rather  more  than  7  inches,  for 

The  square  of  5  is  equal  to  25,  and  as  the  stress  on  a  is 
double,  25  multiplied  by  2,  is  equal  to  50,  the  square  root 
of  which  is  7'07- 

III. 

351.  But  we  may  suppose  it  prudent  to  take  a  standard 
somewhat  larger  than  the  side  of  b,  for  all  the  additional 


CHAP.  111.3  ON   COUPLINGS.  289 

weight  of  the  ports  would  never  be  felt  hurtful  in  prac- 
tiee;  let  us  suppose  6  inches  therefore  a  proper  stand- 
ud  for  eight  horses'  power,  at  forty  revolutions  per 
minute. 

Then  the  square  of  6  is  equal  to  36,  and  36  mul- 
tiplied by  2  is  equal  to  ^%  the  square  root  of  which  is 
S*5  nearly,  or  the  size  which  the  coupling  a  ought  to  have 
had. 

IV. 

352.  In  Observations  II.  and  III.  cases  have  been 
Donsidered  in  which  the  revolutions  per  minute  were 
both  the  same ;  but  we  conceive  that  velocity  must  mate- 
rially affect  durability,  because  the  grinding  or  wearing, 
where  there  is  any  play,  must  be  increased  by  an  increase 
of  velocity. 

v. 

353.  With  respect  to  durability,  couplings  may  be  con- 
sidered under  two  distinct  classes. 

1.  Those  having  boxes. 

2.  Those  without  boxes,  having  legs,  such  as  glands, 

&C. 

The  durability  of  the  latter  class  wiU  probably  increase, 
all  other  circumstances  being  equal,  nearly  in  the  ratio  of 
the  distance  of  the  parts  of  pressure  from  the  centre 
of  motion.  For,  in  glands,  the  pressure  is  commonly 
Hmfined  to  a  small  space  compared  with  that  of  coupling- 
boxes. 

SUPPLEMENTARY  OBSERVATIONS. 

I. 

354.  The  durability  of  couplings  depends  upon  so  many 
circimiBtanoes,  that  it  is  difficult  to  form  general  rules  with 


900  ON   COUPLINGS.  [CHAP.UL 

regard  to  them.  The  two  drcumstonces  abeady  men- 
tioned (Art  350)  are  important,  but  others  merit  at  least 
equal  consideration ;  the  angle  which  the  surface  makes 
with  the  direction  of  the  motion.  Thus  a  square  coupling 
will  be  more  durable  than  an  octagon  of  the  same  size,  be- 
cause the  acting  surface  makes  a  greater  angle  with  the 
tangent  to  the  circle  in  which  the  acting  part  moves.  A 
right  angle  to  the  tangent,  or  the  radius,  wiU  be  the  mmi- 
mum. 

II. 

355.  The  durability  of  couplings  depends  greatly  upon 
the  accuracy  of  the  execution.  For  instance,  a  well-fitted 
square  coupling  will  have  one  fourth,  or  perhaps  one  third 
of  the  surface  of  each  side  acting.  But  if  fitted  as  is 
common  in  practice,  they  will  have  little  more  than  the 
comers  acting.     One  half  of  each  side  is  the  maximum. 


III. 


356.  As  square  couplings  of  a  large  size  are  commonly 
fitted,  there  is  so  little  of  the  surface  acting,  that  we  sup- 
pose their  durability  will  be  nearly  as  the  length  of  the 
box,  multiplied  by  the  velocity  of  the  corners.  But  if 
fitted  as  they  ought  to  be,  their  durability  will  be  as  the 
rectangle  of  the  acting  parts  multiplied  by  the  velocity. 
Or,  to  simplify  the  case,  (as  the  ratio  will  be  the  same,)  as 
the  rectangle  of  the  side,  multiplied  by  the  velocity. 

These  observations  may  perhaps  suggest  matter  of  useful 
practical  reflection  to  the  considerate  millwright 


ESSAY    IV. 


ON    THE   METHODS   OF 


DISENGAGING   AND   KE-ENGAGING    MACHINERY, 


WHILE  IN  MOTION. 


INTRODUCTION. 

The  subject  of  this  Essay  is  so  intimately  connected  with 
that  of  Essay  III.  on  Couplings,  that  in  some  cases  they 
are  really  blended ;  and  while  a  contrivance  is  employed 
for  disengaging  and  re-engaging  machinery,  it  also  serves 
as  a  longitudinal  connexion  of  shafts. 

In  viewing  for  the  first  time  a  cotton-mill,  few  objects 
attract  more  attention,  or  excite  more  pleasing  surprise, 
than  the  facility  with  which  even  children  stop  or  set 
agoing  particular  parts  of  the  mechanism  separately  from 
the  rest  But  however  curious  such  things  may  be  to  in- 
spect, it  is,  perhaps,  no  easy  task,  on  paper,  to  render 
them  interesting  to  the  reader.  Their  utility,  however, 
in  practical  mechanics,  should  stimulate  the  inquirer  to 
examine  with  attention  a  subject  where  ornament  of  style 
is  inadmissible,  and  where  perspicuity  alone  should  be  at- 
temptedy  and  which  it  is  perhaps  not  always  easy  to  attain. 


292  OF   DISENGAGING   AND  [eSSAY  lY. 

The  plan  followed  in  this  Essay  is  similar  to  that  in 
Essay  III.,  that  is  to  say,  in  describing  each  method,  and 
making  separate  observations ;  and  the  prefaitory  observa- 
tions to  that  essay  are  equally  applicable  here. 

Great  credit  is  due  to  Buchanan  in  bringing  into  one 
point  of  view  so  many  inventions,  which  will  not  fail  to  be 
useful  to  the  mechanic,  by  enabling  him  more  easily  to 
compare  them  one  with  another,  and  more  readily  to  se- 
lect such  as  may  be  best  adapted  to  his  purpose. 

357*  From  what  has  been  said  respecting  couplings,  it 
may  easily  be  understood  how  shafts  may  be  disconnected 
when  at  rest  But  many  cases  in  practice  require  that 
particular  parts  of  a  miU  must  be  stopped,  or  set  agomg*, 
without  stopping  or  making  any  sensible  alteration  on  the 
motion  of  the  rest  of  the  machinery.  In  cotton-mills,  for 
instance,  this  becomes  absolutely  necessary ;  and  there  can 
be  no  doubt,  that  necessity,  in  this  case,  has  given  rise  to 
many  most  ingenious  contrivances.  Previously,  however, 
to  the  invention  of  cotton-mills,  there  were  contrivances  for 
this  purpose  in  use ;  such,  for  example,  as  the  sack-tackle 
in  corn-mills. 

358.  In  order  to  assist  us  in  forming  a  judgment  of  the 
comparative  merits  of  such  improvements,  it  may  be  pro- 
per to  bear  in  mind,  a  tendency  attached  to  all  matter 
which  is  intimately  connected  with  practical  mechanics, 
but  on  which  daily  experience  shews  too  little  attention  is 
bestowed  in  the  construction  of  machinery. 

The  tendency  here  alluded  to  has,  by  philosophers, 
been  called  inertiafj  (or  more  frequently,  though  with  less 

*  When  any  particular  part  of  machinery  is  set  agoing,  it  is  said  among 
workmen  to  be  set  on^  or  put  in  pear;  when  stopped,  set  off'  or  ptU  out  of  gear. 

t  "  A  tendency  to  preserve  in  a  state  of  rest  or  unifonn  rectilinear  mo- 
tion, is  a  property  attached  to  all  matter,  and  may  be  considered  as  pro- 
portional to  the  mass  or  weight  of  a  body."  (Young's  Nat.  Phil.  Vol.  I. 
p.  51.) 


WAY  IV.] 


RE-ENGAGING    MACHINERY. 


propriety,  vis  inertia,*)  by  which  is  meant,  the  tendency 
which  every  piece  of  matter,  when  at  rest,  has  to  remain 
i  rest ;  and,  when  in  motion,  to  continue  in  motion.  In 
fcer  words,  the  impossibility  of  instantaneoitsly  producing 
lotion  in  a  body,  or  of  tmtnntaneousiy  stopping  a  body  in 
It  is  this  tendency,  therefore,  that  occasions 
J  violent  shocks  in  attempting  to  set  bodies  suddenly 
jnto  motion  ;  and  those  shocks,  besides  tending  to  destroy 
the  machine,  occasion  a  very  great  loss  of  power.  Some 
^urt  of  the  machine  must  break,  or  at  least  yield  to  this 
^■tent  law. 

^V359<  The  practical  mechanic,  and  perhaps  also  the  phi- 
loeopher,  will  find  it  some  advantage  to  dismiss  the  term 
inertia  from  the  place  it  occupies  in  science.  Wlien  pro- 
perly understood,  it  simply  indicates  that  matter  never 
changes  its  state,  unless  there  be  a  change  in  the  power  or 
powers  acting  upon  it. 

If  you  consider  inertia  as  a  power,  it  must  be  identical 
with  the  power  of  gravity  in  one  case,  with  the  momentum 
of  a  body  in  another,  with  the  power  of  magnetism  in  a 
third,  with  friction  in  a  fourth,  and  so  on  throughout  the 
whole  of  the  powers  in  nature ;  but  many  writers  write  as 
though  it  were  a  real  power,  and  distinct  from  all  these, 
aad  consequently  lead  their  readers  into  incorrect  notions 

tthe  subject. 
It  13  easily  proved,  that  when  a  body  is  struck  by  an- 
ler  in  motion,  some  time  is  occupied  in  communicating 
the  motion  from  the  point  struck  to  the  other  parts  of  the 
body  ;  and  therefore,  if  the  parts  receiving  the  blow  have 
fc  sufficient  elasticity  and  cohesive  power  to  destroy  the 
ble  momentum  of  the  striking  body  till  the  motion  bo 

'    Vi»  inertia  is  defined  by  Newton,  (Def.  3.  Book  I.)  lo  be  o  power 
*U)te<l  in  nil  matter,  by  which  it  resists  any  change  endeavoured  to  bs 

is  in  its  BtBto;  tliut  is,  by  which  it  becomes  dillicnU  to  ultcr  its  stale, 

nr  of  rett  or  motion. 


994f  OF  DI8BN6AOINO   AND  [bSSAT  IV. 

transmitted  to  the  centre  of  rotation,  it  must  necesssnly 
break  the  body  struck*. 

Hence  any  part  of  a  machine  intended  to  be  acted  upon 
suddenly  by  a  moving  power  should  be  strong  but  yielding, 
or  it  may  be  capable  of  sliding  with  much  friction  on  the 
body  to  be  put  into  motion. 

360.  Let  us  illustrate  this  by  example.  Throwing  a  whed 
into  gectTf  very  often  occasions  the  breaking  of  the  teeth, 
whereas  a  pulley  is  generally  put  in  motion  by  a  belt  so 
g]^adually,  that  no  part  of  the  machinery  can  receive  any 
injury.  In  the  first  case,  the  wheel  being  firmly  fixed  on 
its  shaft,  can  yield  little  to  inertia^  (the  moving  foroe.) 
In  the  last,  there  is  the  elasticity  of  the  belt,  as  well  as  the 
liberty  of  slipping  on  the  pulley  to  yield  to  the  inertia^ 
(moving  force,)  till  by  its  friction,  the  belt  gradually 
brings  it  into  motion. 

361.  This  subject  naturally  divides  itself  into  the  two 
following  sections : 

I.  Of  Methods  used  when  motion  is  communicated  hj 
means  of  bands,  belts  or  chains. 

II.  Of  Methods  when  motion  is  communicated  by  means 
of  wheel-work. 

METHOD  I. 

THB   SLIDING   PULLBY. Fig.  1,  Plate  IX. 

362.  The  sliding  pulley  is  one  of  the  oldest  contrivances 
for  disengaging  and  re-engaging  a  machine  moved  by  a 
belt  or  band.  The  pulley  p,  which  is  driven  by  the  mill- 
work,  and  gives  motion  to  the  machine,  is  not  fixed  dead 
on  the  axle  a  b,  but  has  a  hollow  cylindrical  bush  made  of 
met^,  accurately  fitted  to  the  axle,  so  that  it  may  revolve 
easily  upon  it,  and  slide  a  little  backward  and  forward. 

*  Tlie  rales  for  estimating  the  power  of  materials  under  these  circum- 
stance may  bo  found  in  Tredgold's  Practical  Essajs  on  the  Strength  of 
Cast  Iron,  Sect  VII. 


ISAV  IV.] 


RE-ENCAGING    MACHINERY. 


295 


:  order  to  make  the  pulley  p  carry  round  the  axle  a  b, 
!  is  a  cross  piece  or  gland  de  firmly  fixed  to  it.  On 
nc  side  of  the  pulley  which  is  toward  the  cross,  there  is 
i*ne  or  more  teeth  T  ;  when  the  puller  p  is  moved  toward 
the  cross  de,  the  teeth  lay  hold  of  it,  and  thereby  carry 
round  the  axle.  By  sliding  it  backward,  though  the  pul- 
ley still  continues  in  motion,  the  teeth  t  are  disengaged 
from  the  cross  de,  of  course  the  axle  stops,  and  with  it 
the  machine  to  which  it  gives  motion. 

The  bush  c,  of  the  pulley,  projects  a  little  upon  the  side 

opposite  the  teeth,  and  has  a  groove  cut  in  its  outside. 

luto  this  groove,  a  lever  fg  lies,  but  so  as  not  to  prevent 

s  motion  of  the  bush.     By  moving  the  lever  fo  the  bush 

[  moved  along  with  it,  and  thus  serves  to  disengage  or 

age  the  pulley. 


OBSERVATIONS. 

IS63.  For  a  long  time  this  contrivance  was  applied  to 
I  carding  machines.  The  shock,  however,  proceed- 
f  from  inertia,  (Art.  35S,)  occasioned  by  the  teeth  strik- 
r  against  the  fixed  cross  in  setting  the  machine  agoing, 
was  verj"  great,  and  apt  to  shake  the  cylinders  loose  from 
their  axles.  This  is  a  very  great  defect,  and  for  such  pur- 
poses as  have  been  mentioned  other  methods  more  perfect 
are  now  used. 

^B  864-.  The  pulley  or  hinder  a,  is  kept  in  motion  by  a 
^Telt.     It  has  a  bush  fitted  to  the  upright  axle  uc,  and 
runs  always  at  the  same  height. 

On  iia  upper  side,  the  binder  a  has  t«eth  t,  t  projecting 
ward ;  above  is  a  pulley,  or  other  piece  of  wood  or  metal 


METHOD  II. 


THE   BAYOKHT.— Fig.  2. 


296  OF   DISENGAGING   AND  [eSSAY  IV. 

DE,  fixed  to  the  shafit  bc,  through  holes  in  which  passes 
the  legs  F,  G  of  the  hayonet*. 

The  bayonet  is  represented  by  Fig.  2,  No.  3.  It  is 
merely  a  piece  of  metal  with  two  or  more  legs,  which  pro- 
ject downward.  The  part  h  is  made  hollow  to  receive  the 
shaft,  and  has  a  groove  into  which  ik  fits,  and  the  lifter 
moves  it  upward  or  downward,  serving  the  same  purpose 
as  the  lever  fg.  Fig  1. 

When  the  bayonet  is  dropped  downward,  it  lays  hold  of 
the  teeth  t  of  the  binder  a,  which  immediately  carries  it 
round,  and  this  gives  motion  to  the  axle  bc;  all  that  is 
required  to  stop  it,  is  merely  by  hand  to  raise  the  lifter  ik, 
which  has  a  spring  or  catch  to  keep  it  in  that  situation, 
until  again  dropped  by  the  hand  of  the  attendant 


OBSERVATIONS. 

365.  This  contrivance  is  more  perfect  than  the  sliding 
pulley,  Fig.  1 ;  because  the  pulley  does  not  require  to  be 
shifted  in  the  direction  of  the  shaft,  which  is  often  incon- 
venient, and  requires  more  space.  Fig.  2.  has  however 
the  same  defect  as  Fig.  1,  of  producing  a  shock  at  the  in- 
stant when  the  machine  is  put  in  gear. 

This  method  is  still  much  used  in  cotton-mills.  The 
figure  represents  one  of  the  upright  shafts  of  a  spinning 

*  Instead  of  the  legs  of  the  bayonet  passing  through  holes,  as  here  de- 
scribed, it  is  sometimes  made  square  in  the  inside,  so  as  to  fit  a  square  part 
of  the  shaft ;  which  square  answers  the  purpose  of  the  piece  of  wood  or 
metal,  viz.  that  of  carrying  round  the  bayonet  along  with  the  shaft.  The 
bayonet  has  sometimes  one  leg  only ;  on  other  occasions  it  is  made  like  a 
face-wheel,  having  a  number  of  projecting  teeth  or  legs.  Indeed,  it  is  almost 
obvious,  that  all  or  most  of  the  contrivances  which  are  mentioned  in  this 
Essay,  must  be  variously  modified,  according  to  circumstances,  or  the  fancy 
of  the  artisan.  I  have  only  given  the  way  in  which  they  are  most  generally 
applied.  To  give  all  the  varieties  would  be  an  almost  endless,  and,  per- 
haps, useless  labour. 


METHOD  III. 


OP  TSB   LOCK   PULLEY,— Fig.   : 


Bay  IV.3  RE-ENGAGING    MACHINEIIY.  297 

frame.     The  same  contrivance  is  often  applied  to  horizon- 
tal shafts. 

^HiS66.  This  method  is  somewhat  similar  to  the  last  de- 
^^cribed,  but  instead  of  a  bayonet,  it  has  a  lock  at  a,  the 
bolt  of  which  b  lays  hold  of  the  cross  cc,  which  is  fixed  to 
^the  axle.  The  pulley,  when  disengaged,  runs  on  a  bush, 
^b  is  the  key,  which  is  turned  by  means  of  a  stop,  which  la 
^Bade  to  touch  a  cross  part  on  the  end  of  the  key.  The 
^Htrning  of  the  key  throws  back  the  bolt,  and  so  unlocks 
^Kid  disengages  the  pulley. 

367-  This  contrivance  is  less  simple  than  the  bayonet. 
Nor  indeed  has  the  lock  pulley  ever  been  generally  adopted ; 
a  proof  that  it  has  not  been  found  of  much  real  use.  It  was 
used  at  Manchester  about  the  time  that  machines,  called 
mules,  for  spinning  cotton,  were  begun  to  be  changed  from 
being  moved  by  hand,  to  receive  their  motion  (as  is  now 
leral)  by  power. 


OBSERVATIONS. 


TBB  FAST  AND  LOOSE  PULLBV. — Fig.  4,  Plate  X. 


368.  The  pulley  b  is  fixed  on  the  axle  a,  and  the  pulley 
c,  having  a  bush,  is  loose.  The  belt  or  band  which  con- 
veys the  motion,  may,  at  pleasure,  either  by  hand,  or  by  a 
leTer,  be  shifted  from  the  one  pulley  to  the  other.     WTien 

^running  on  the  loose  pulley  c,  the  axle  stands  still ;  when 

^■h  the  fast  pulley  b,  the  axle  moves. 


S98  OF   DISENGAGING   AND  [eSSAT  IV. 

No.  1,  represents  the  pulleys  as  adapted  to  a  belt. 

No.  2,  represents  the  pulleys  as  made  for  a  rope.  The 
inner  ledges  are  made  low,  in  order  that  the  rope  may 
slide  with  the  greater  facility  from  the  one  to  the  other. 


OBSERVATIONS. 

369*  It  may  be  proper  here  to  mention,  that  in  order  to 
make  a  belt  run  properly  on  a  pulley,  it  is  necessary  to 
have  the  rim  of  the  pulley  a  little  rounded  or  swelled  in 
the  middle.  The  belt  always  inclines  to  that  part  of  the 
pulley  which  is  of  greatest  diameter.  This  curious  pro- 
perty is  found  of  great  practical  use.  Until  this  property 
was  known,  it  was  found  very  troublesome  to  get  belts  pre- 
vented from  slipping  off  the  pulleys. 

370.  This  contrivance  of  the  fast  and  loose  pulleys,  i« 
remarkable  for  its  beautiful  simplicity.  It  is  attended  with 
no  shock,  and  is  perhaps  the  most  perfect  thing  yet  in- 
vented for  the  purpose,  in  all  cases  where  it  can  be  applied 

Its  application  in  cotton-mills  is  now  general.  Never, 
until  it  was  applied,  were  the  spinning  mules  found  to  give 
satisfaction,  when  moved  by  power.  They  are  now,  how- 
ever, almost  exclusively  wrought  by  power.  This  im- 
provement has  not  only  produced  a  great  saving  of  labour, 
but  has  been  attended  with  another  very  pleasing  effect 
It  has  rendered  the  employment  of  the  spinner  more  health- 
ful than  when  moved  by  hand.  The  severity  of  the  hand- 
labour  in  a  warm  and  confined  atmosphere,  had  a  very  un- 
favourable effect  on  the  constitution. 


METHOD  V.-Fig.  5. 

371.  The  undermost  axle  ab  is  kept  by  the  mill  in  con- 
stant motion.     The  uppermost  axle  c  d  has  a  bridge  at  d, 


ESSAY   IV.]  ItE-ENGAGING    MACHINERY.  399 

to  the  form  of  a  lever,  in  order  to  raise  or  lower  it  at 
pleasure.  WTien  the  end  d  of  the  shaft  ia  raised  by  means 
of  the  bridge,  the  chain  or  rope  e  f  is  tightened,  which 
causes  the  axle  b  to  move.  But  when  d  is  lowered,  the 
chain  falls  slack  below  the  pulley,  on  the  lowermost  shaft  a, 

1}  thus  disengages  the  shafts. 
Er 


OBSERVATIONS. 


'2.  This  very  simple  contrivance  is  used  iu  some  parts 
England,  in  corn-mills,  for  the  sack-tackle.  It  is  evi- 
dently applicable  to  many  other  purposes  ;  and  it  is  obvious, 
that  E  F  may  he  either  a  belt,  a  rope,  or  a  chain.  A  chain 
is  commonly  used  when  applied  as  a  sack-tackle.  In  that 
case,  the  rope  or  chain,  for  taking  up  the  sacks,  is  wound 
id  the  barrel  on  the  axle  c  d. 


SECTION  H. 

I  OP   TBE    MHTH0P8   «3ED  WnEN   MOTION   IS  COWVEYEB  BV    UEANS  Of 

73.  When  a  mill  is  in  motion,  we  may,  with  perfect 
■ty.  lift  a  wheel  out  of  gear ;  but  in  throwing  wheels 

V  gear,  when  a  mill  is  in  motion,  there  is  great  risk  of 
ing  the  teeth ;  it  is,  nevertheless,  often  done.  Cot- 
I  spinners  know  fcojn  experience,  that  the  risk  of  break- 
r  the  teeth  is  much  lessened  when  the  wheel  to  be  thrown 
info  gear  is  previously  set  in  motion  by  the  hand.  This 
they  do  by  pulling  the  shaft  round  by  the  belt.  The  reason 
bvious.  The  inertia  is  lessened  by  this  motion  given 
iic  hand  to  the  wheel  which  is  to  be  thrown  into  gear. 

METHOD  Vr.— Fig.  6. 

74.  Wheels  are  often  disengaged  and  re-engaged  by 


300  OF   DISENGAGING   AND  [eSSAT  IT. 

means  of  one  of  the  bridges,  ab,  which  carries  the  end  of 
the  shaft  nearest  the  wheel,  acting  as  a  lever,  haying  its 
jvlcTum  A  at  one  end.  The  other  end  b  is  moveable  in 
one  direction,  and  is  raised  or  allowed  to  fall,  by  turning 
the  key  c  d  ;  when  out  of  gear  the  bridge  rests  securely  on 
the  end  of  the  key,  as  represented  in  No.  2,  Plate  XL 
No.  1,  represents  the  wheels  in  gear. 

Sometimes  the  bridge  is  raised  immediately  by  hand, 
sometimes  by  a  rope  and  pulley ;  in  these  cases  it  is  held 
out  of  gear,  by  interposing  a  wedge  instead  of  the  key,  or 
by  a  catch  under  the  end  of  the  bridge. 

OBSERVATIONS. 

375.  This  mode  of  disengaging  wheels  is  sometimes  in- 
convenient for  want  of  room;  it  has,  however,  the  ad- 
vantage of  having  the  wheels  firmly  fixed  on  the  shafts. 
In  falling  into  gear  the  ends  of  the  teeth  are  apt  to  strike 
against  one  another,  and  there  is  often  a  violent  shock 
from  the  inertia. 

In  the  figures,  the  wheels  are  represented  the  one  above 
the  other ;  but  the  same  contrivance  is  modified,  and  ap- 
plied, when  they  happen  to  be  in  any  other  relative 
situation*. 

METHOD  VII.— Fig.  7,  Plate  XI. 

376.  Instead  of  lifting  wheels  out  of  gear,  in  order  to 
disengage  any  piece  of  mechanism,  one  of  the  wheels,  in- 
stead  of  being  fast  on  the  shaft,  has  a  round  bush  like  a 
loose  pulley,  and  a  clutch,  or  bayonet,  which  connects  it 
with  its  shaft,  in  a  similar  manner  with  the  bayonet  and 
binder.  Fig.  2. 

*  The  motion  of  the  wheel,  as  it  falls  into  gear,  should  be  in  the  same 
direction  as  it  is  afterwards  to  move  in ;  which  will  lessen  the  shock  cod- 
siderahly. 


F 


AY  IV.3  RE-ENGAGING   MACHINERY.  301 

Thus  the  wheel  a,  Fig.  7,  No.  1,  has  a  busb,  and  works 
on  a  round  part  of  the  shaft  b.  The  clutch  d,  may  slide 
on  a  square  part  of  the  same  shaft,  and  is  disengaged  or 
re-engaged  at  ])leasure,  by  means  of  the  lever  ef,  part  of 
which  is  represented  as  cut  off,  in  order  to  shew  the  groove 
man  the  clutch  into  which  it  acts. 

^L  No.  2,  is  an  elevated  section  of  the  lever  ef,  shewing  its 
^^■Dnexion  with  the  clutch  d,  by  means  of  two  iron  pins 
^^Bg,  screwed  into  two  slips  of  iron  H,  H,  which  fit  the  groove, 
^^Bd  are  not  affected  by  any  oblique  position  the  lever  may 
have. 

OBSERVATIONS. 

S77.  This  contrivance  is  sometimes  made  more  con- 
venient than  is  shewn  in  Fig.  7>  by  occupying  less  space ; 
and  although  it  is  subject  to  a  shock  at  setting  on,  yet  it 
is  not  so  liable  to  break  the  teeth  as  in  Method  VI. 

METHOD  VIII. 

I   CLUTCH. — Fig.   8. 


S78.  This  figure  represents  the  friction  clutch ;  it  dif. 
fers  from  the  bayonet  (Method  II.)  in  this  respect,  that 
instead  of  striking  on  a  fast  cross,  the  bayonet  or  clutch 
lavs  hold  of  the  ears  of  a  screwed  hoop,  which  embraces  a 
kind  of  drum. 

AB  represents  part  of  a  shaft  kept  in  motion  by  the 
mill;  CDE  a  bayonet,  which  either  slips  on  a  square  part 
of  the  shaft  ab.  or  passes  through  the  arms  of  a  cross  I'q, 
(as  represented  in  the  figure,)  which  cross  is  fastened  to 

AB. 

FG  is  part  of  a  shaft  to  be  connected  with  the  shaft  ab  ; 
upon  FG  a  kind  of  drum  or  pulley  11  ik  is  fastened;  this 
pulley  has  ledges  to  keep  the  screwed  hoop  j.mno  steady. 


I 


302  OF  DISEHGAOINO  AND  [BttAl  £?. 

In  setting  on  the  machine,  the  ho(q[)  lmko  is  canied 
round  hy  the  hayonet  or  clutch  ci>£»  and  by  the  fricdon  of 
the  hoop  on  the  drum  hik,  brings  it  into  motion,  in  tin 
some  easy  and  gradual  manner  that  a  belt  does  a  machiiie 
driven  by  a  pulley. 

The  hoop,  as  represented  in  No.  3»  Plate  XIL,  fixim 
acting  more  perfectly  as  a  spring,  is  found  to  answer  better 
in  practice  than  that  represented  in  No.  1.  and  No.  2. 


OBSERVATIONS. 

379*  There  is  a  great  deal  of  beauty  in  this  ingenious 
contrivance,  it  may  be  appUed  to  the  largest  machinery,  and 
variously  modified  according  to  circumstances*.  It  is  ob- 
vious that  it  prevents  all  the  unpleasant  and  hurtful  shock 
so  common  in  throwing  heavy  machinery  into  gear. 

It  may  also  be  the  means  of  saving  lives,  for  should  a 
person's  clothes  be  laid  hold  of  by  the  wheel- work,  in  most 
cases  the  hoops  would  slip  on  the  drum,  and  allow  that 
part  of  the  machinery  to  stop,  without  sensibly  altering  the 
general  motion  of  the  mill.  It  may  in  this  way,  too,  pre- 
vent injury  to  the  mill  itself,  which  might  arise  from  belts 
getting  foul,  or  from  chips  falling  in  among  the  teeth  of 
the  wheel- work. 

The  friction  clutch  has  been  lately  applied  to  frames  for 
spinning  flax,  and  it  seems  to  me,  might  also,  with  great 
advantage,  be  applied  to  frames  for  spinning  cotton  water- 
twist. 

METHOD  IX. 
THE  FRICTION  C0NB8.— Fig.  9,  Plate  XIL 

380.  This  contrivance  is  similar  in  its  principles  and  ef- 
fects to  the  friction  clutch. 

*  Thus,  for  instance,  it  may  be  applied  to  the  wheel  and  clutch. — Fig.  7* 


ESSAY  IV.]  nE-ENGAGlNC    MACHINERY.  303 

On  the  shaft  a  (kept  m  motion  by  the  mill)  there  is 
fixed  a  hollow  cone  h  ;  on  the  shaft  B  is  another  cone  e, 
(he  external  part  of  which  fits  the  internal  part  of  h  ;  e  is, 
liowever,  moveable  like  a  bayonet  on  a  square  part  of  the 
f;haft  B,  and  may  be  moved  outward  and  mward  also  hke  a 
bayonet,  by  a  lever. 

MTien  e  is  moved  forward,  it  rubs  on  the  hollow  part  of 
n,  and  by  iriction,  like  the  triction  clutch,  gradually  brings 
the  machine  connected  with  a  into  motion. 


i 


OBSERVATIONS. 


WHBBLS   ACTING   BV    FRICTION, — Fig.    10. 


S81 .  Sometimes  there  is  added  a  bayonet,  passing  through 
a  hollow  cone,  wliich,  should  occasion  require,  gives  liberty 
to  lock  the  shafts  quite  fast  into  one  another. 

The  friction  cones  are  sometimes  applied  to  sack-tackles  ; 
one  of  them  on  this  construction  may  be  seen  at  Meux's 
bfewery. 

^^fdSQ.  Sometimes  wheels  are  made  to  act  without  teeth,  as 
represented  in  the  figure ;  they  move  one  another  by  con- 
tact, having  their  circumferences  generally  made  of  end 
grain  of  wood,  which  may  indeed  be  considered  as  forming 
indefinitely  small  teeth. 

It  is  evident,  that  wheels  of  this  kind  may  work  with 
Uittlc  noise,  and  be  put  into  gear,  and  bear  against  one 
Rttiother  without  risk  of  damage.     They  are  commonly  dis- 
engaged and  re-engaged  by  a  bridge,  acting  as  a  lever,  on 
simitar  principles  as  described  in  the  case  of  Fig.  6. 

»  OBSERVATIONS. 

383.  This  species  of  wheel-work  has  been  used  with 
;.  .od  effect  in  machinery  for  raising  coal ;  it  is  also  used, 


304  OF  DISENGAGING  AND  [£SSAT  IV. 

in  some  cases,  in  cotton-mills ;  and  has,  for  a  nmnber  of 
years,  been  employed  in  a  saw-mill  by  Taylor  of  South- 
ampton, the  principle  and  method  of  which  in  transmittmg 
mechanic  power  certainly  deserves  attention. 

METHOD  XL— Pig.  II. 

384.  The  figure  represents  another  application  of  fric- 
tion in  transmitting  mechanic  force,  but  instead  of  the 
friction  being  on  the  hem  of  the  wheel,  as  in  Fig.  10,  it  is 
here  applied  to  the  sides  of  the  wheel. 

The  mode  here  represented,  is  successfully  put  in  prac- 
tice, in  a  tackle  for  raising  and  lowering  sacks  in  a  r^ 
spectable  brewhouse  in  London,  a  is  the  axle  which  gira 
motion  to  the  tackle,  upon  which  is  the  friction  wheel  b  ; 
upon  the  axis  a  are  the  friction  wheels  c,  d,  and  the  roller 

E,  round  which  the  rope  winds.  The  end  of  the  axis  a  runs 
in  a  socket  in  the  end  of  the  axis  a  ;  the  other  end,  in  die 
brass,  is  in  the  post^  In  raising  the  sacks,  the  wheel  c 
is  kept  fast  against  the  wheel  b,  by  the  lever  h  and  catches  cc. 

For  lowering  sacks,  the  wheel  d  is  kept  against  the  wheel 

F,  by  the  lever  h  and  weight  d  going  over  the  pulley  e,  and 
a  man  holding  the  line  g  in  his  hand,  makes  the  friction 
of  D  upon  E  more  or  less,  as  is  necessary. 

OBSERVATIONS. 

385.  This  is  a  very  ingenious  and  simple  machine,  and 
as  its  principles  might  be  applied  in  other  cases,  it  is  well 
worth  the  attention  of  the  millwright. 

METHOD  XII. 

SELF-DISBNOAOINO   COUPLING.— Fig.   12. 

386.  A  represents  a  shaft,  kept  in  motion  by  the  mill ; 
B  c  a  cast  iron  wheel  fast  on  the  shaft  a,  having  four  pro- 


IBSAY  IV.]  RE-ENCAGING    MACHINERY. 


305 


f'jecting  teeth,  d, d,  &c.,  of  wrought  iron;  ef,  is  another 
lilar  wheel,  with  similar  teeth,  g,  g,  &c.  but  is  loose  on 
}  shaft  H,  and  is  made  to  slide  on  it,  and  to  act  as  a  kind 
f  bayonet.  (Method  II.)  The  teeth  project  obliquely,  as 
lay  be  seen  in  the  figure. 
I  When  the  coupling  is  engaged,  the  teeth  lay  hold  of  one 
lother,  and  the  shaft  h,  is,  by  their  means,  carried  round 
nth  the  shaft  a,  but  when  any  extraordinary  stress  comes 
.  the  shaft  h,  the  pressure  on  the  oblique  teeth  forces 
tck  the  bayonet  e  f,  and  disengages  the  coupling,  i  k  l, 
la  bended  lever,  having  its  fulcrum  at  k,  the  bayonet  is 
pt  forward  by  the  weight  of  the  part  m  k,  of  the  lever, 
r  the  ordinary  stress  on  h. 
■When  the  bayonet  ef,  is  forced  back,  the  lever  is  held 
I  by  a  catch,  until  the  coupling  is  re-engaged  by  the  hand 
I  the  attendant.  The  coupling  is  represented  in  the  figure 
I  disengaged'. 

*  In  order  to  succeed  in  producing  this  effect,  the  angle  bao  must  be 

k  aanewhat  greater  than  would  cause  the  surfaces  to  slide  upon  one  another, 

irheo  acted  upou  by  a  pressure  in  the  direction  de,  perpendicular  to  a  c. 

Then,  when  the  machine  is  in  motion  they  would  actually  slide  apart,  were 

it  not  for  the  frictioD  on  the  shaft,  and  the  weight  of  the  lever. 

If  the  angle  ba  c  be  less  than  the  angle  which  would  cause  the  bodies  to 
■IHe,  the  coupling  would  not  disengage  itself  by  any  force  whatever. 

Accordiiig  to  Coulomb's  experiments,  (see  Brewster's  Additious  to  Fer- 
guson's Lectures,  vol.  ii.  p,  155,)  the  friction  of  iron  on  iron  is  about  one 
fourth  of  the  pressure,  hence  the  angle  bac  should  be  greater  than  15 
degrees;  otherwise  the  coupling  will  not  disengage. 


^ 


306  OF   DISENGAGING,   ETO,    MACHINERY.      [SSSAT IV. 

OBSERVATIONS. 

387.  This  coupling,  as  it  prevents  accidents  from  any 
sudden  stress,  is  found  very  useful  where  turning  lathes 
are  driven  by  wheel-work. 

Some  good  instances  of  self-disengaging  apparatus  may 
be  seen  in  looms  driven  by  power.  Respecting  these  ma- 
chines, the  reader  is  referred  to  Duncan's  "  Essays  on 
Weaving,"  as  also  to  the  Edinburgh  Encyclopedia*. 

*  It  will  always  be  found,  that,  in  engaging  by  wheels,  the  teeth  will  be 
less  liable  to  be  stripped  in  small  wheels  than  in  laige  ones ;  becaiue  in 
small  wheels  the  stroke  will  be  made  with  a  less  d^;ree  of  velocity,  ind 
also  a  small  wheel  requires  less  force  to  put  it  in  motion ;  hence  a  small  and 
light  wheel  with  strong  teeth  will  seldomer  fiul  ihan  a  heavy  one.  Peikpi 
the  best  arrangement  will  be  when  the  wheel  in  constant  motion  is  sbuS^ 
and  that  to  be  occasionally  put  in  motion  a  laiger  one,  with  elastic  aniii 
These  elastic  arms  mig^t  be  made  in  the  manner  of  coodi  q^rings. 


FtHl 


ESSAY    V. 


MECHANISM 


EQUALIZING  THE  MOTION  OF  MILLS,  DENOMINATED 
LIFT-TENTERS,  ENGINE  GOVERNORS,  AND  WATER- 
WHEEL  GOVERNORS. 


INTRODUCTION. 


:I3  Efisaj  relates  to  machinery  not  less  curious  in  its 
construction  than  useful  in  practice ;  and  as  some  of  the 
apparatus  is  intimately  connected  with  water-wheels,  the 
papers  which  are  suhjoined,  containing  an  account  of  some 
experiments"  and  observations  on  their  velocity,  may  not 
be  unacceptable  to  the  reader,  who  will  observe  that  when 
a  part  of  the  machinery  of  a  mill  js  suddenly  stopped,  or 
euddenly  set  a-going,  and  the  moving  power  remains  the 
same,  an  alteration  in  the  velocity  of  the  mill  wUl  take 
place;  it  will  move  faster  or  slower.  Every  macliine 
having  a  certain  velocity  at  which  it  will  work  at  greater 
advantage  than  at  any  other  speed,  the  change  of  velocity 
arising  from  the  above  cause,  is  in  all  cases  a  disadvantage, 
and  in  delicate  operations  exceedingly  hurtful.  In  the 
ease  of  a  cotton-mill,  for  instance,  which  is  calculated  to 
move  the  spindles  at  a  certain  rate,  if  from  any  cause  the 

*  An  account  of  tbeee  experimenta  was  ori^&Uy  pulilisbed  in  the  10th 
e  of  the  PLilosophicnl  Magosine,  p.  ITS. 


308  ON   EQUALIZING   THE  [eSSAT  V. 

Telocity  is  much  increased^  a  loss  of  work  immediately  takes 
place,  and  an  increase  of  waste  from  the  breaking  of  the 
threads,  &c. ;  on  the  other  hand,  there  must  be  an  evident 
loss  from  the  knachinery  moving  too  slow. 

388.  In  steam-engines  this  evil  is  remedied  by  a  c(m- 
triyance  called  a  governor.  (Plate  XIII.  Kg.  1.) — "Two 
baUs  are  fixed  to  the  ends  of  rods,  in  continual  revolution, 
and  as  soon  as  the  motion  becomes  a  little  too  rapid,  the 
balls  rise  considerably,''  and,  by  the  intervention  of  a  lever, 
act  upon  a  throtUe-valoe^ ^  which  diminishes  the  quantity 
of  steam  admitted,  and  of  course  serves  to  make  the  motion 
less  rapid. 

SECTION  I. 

THE  8TRAM-BN0INB  GOVBBNOB. — ¥\g.  1. 

389.  IK  represents  a  spindle  kept  in  motion  by  the 
enginet ;  a,  b  the  centrifugal  balls ;  c a  and  cb  the  rods 
by  which  the  balls  are  suspended.  These  rods  cross 
one  another,  and  pass  through  the  middle  of  the  spindle  at 
ۥ  There  is  a  round  pin  put  through  the  spindle  and  the 
rods  at  c,  whicli  serves  as  the  point  of  suspension  for  the 
centrifugal  balls  or  revolving  pendulum.  There  is  a  part 
of  the  spindle  above  c  which  is  square,  and  nicely  polished, 

*  A  thrdttle-vdlve  is  fofrmed  by  a  plate  of  metal,  which  is  fixed  on  a 
spindle  passing  across  the  middle  of  it.  When  the  edge  of  this  round  metal 
plate  is  in  the  direction  of  the  current  of  steam,  the  aperture  is  at  its  great- 
est extent  of  opening ;  and  as  the  plate  becomes  more  oblique  the  opening 
becomes  less,  until  it  is  shut  by  the  plate  being  at  right  angles  to  the 
current. 

The  pressure  on  both  sides  the  spindle  Ibeing  the  same,  this  kind  of  vhItc 
is  opened  or  shut  with  more  ease  than  amy  'other ;  and  is,  therefore,  veiy 
applicable  here.  It  is  not  easy  to  make  it  quite  steam-tight  when  shut,  but 
its  tightness  is  not  of  consequence  in  this  case. 

t  This  motion  is  sometimes  produced  by  a  rope  and  pulleys,  but  wheel- 
work  being  more  certain,  is  much  to  be  preferred. 


AT  v.]  MOTION    OF   MILLS.  309 

t  that  the  piece  of  brass  m  may  slide  easily  up  and  down 

The  piece  of  brass  m  is  round  on  the  outside, 

I  has  an  external  groove  turned  upon  the  upper  end  of 

>  receive  the  lever  n  o,  the  fulcrum  of  which  is  at  p. 

3  piece  of  brass  is  connected  with  the  ball-rods  by  two 

lort  pieces  and  joints  de,  fc. 

I  The  construction  of  steam-engine  governors  sometimes 
Hers  a  little  from  that  now  described  j  but  if  this  par- 
alar  construction  be  imdcrstood,  there  will  be  no  djffi- 
Bty  in  comprehending  any  other  in  use. 


I  390.  When  the  engine  goes  too  fast,  the  balls  fly  oflT 
I  the  spindle,  and  depress  the  end  n  of  the  lever,  which 
tly  shuts  the  throttle-valve,  and  thereby  diminishes  the 
utity  of  steam  admitted  into  the  cylinder;  and,  on  the 
other  hand,  when  the  engine  goes  too  slow,  the  balls  fall 
down  toward  the  spindle,  and  elevate  the  end  n  of  the 
lever,  which  partly  opens  the  throttle-valve,  and  thereby 
increases  the  quantity  of  steam  admitted  into  the  cylinder. 
391-  This  apparatus  being  of  great  practical  use,  and  as 
it  is  applied  to  other  purposes  which  I  am  about  to  de- 
scribe, it  may  be  proper  here  to  give  a  rule  for  the  number 
of  revolutions  which  the  spindle  i  k  ought  to  make  in  pro- 
portion to  the  situation  of  the  balls  with  regard  to  the  cen- 
tre of  their  suspension  c. 

In  order  to  explain  this  rule,  therefore,  it  is  proper  to 
observe,  that  "  there  is  a  great  analogy  between  the  vibra- 
tion of  pendulums  and  the  revolutions  of  balls  suspended 
from  a  fixed  point.     If  a  body  suspended  by  a  thread  re- 
Hmlve  freely  in  a  horizontal  circle,  the  time  of  the  revolution 
^BU  be  the  same  whenever  the  height  of  the  point  of  sus- 
^^ksion  above  the  plane  of  revolution  is  the  same,  what- 

L^ 


310  ON   EQUALIZIKO  THE  [ESaiTV. 

Fig.  %  be  made  to  revolvei  they  will  arrange  tibeoifldyitt  w 
as  to  remain  very  nearly  in  the  same  horicmital  plsn. 
^'  The  time  of  each  revolution  of  the  balls  is  equal  to  tfae 
time  occupied  by  a  double  vibration  of  s  pendolmny  of 
which  the  length  is  equal  to  the  height  of  the  point  of  m- 
pension  above  the  plane  in  which  they  revolva'^* 

392.  Thus,  for  instance^  if  the  height  of  the  point  of 
suspension  d,  Ilg.  S,  above  the  plane  on  which  the  baDs 
revolve,  be  equal  to  the  length  of  a  pendnlnm  whidi  lu 
brates  seconds,  the  balls,  in  that  case,  should  make  SO  le* 
volutions  per  minute. 

393.  If  pendulums  are  of  the  following  lengths,  their 
oscillations  in  one  minute  of  time,  in  Britain,  are  as  fol- 
low: 

BiitiBli  feet  and  inches.  OsdHaiions. 


"Feet    0  .    .     .  Inch  1-565 

0 9782 

3  .     .     •     .     .  3-128 

13 0-512 

52 2-048 


300 

120 

60 

30 

15 


394.  "  Hence  the  oscillations  of  pendulums  are  in  the 
subduplicate  ratio,  or  as  the  square  roots  of  their  lengths ; 
and  the  lengths  of  pendulums  are  in  the  duplicate  ratio, 
or,  as  the  squares  of  their  oscillations.  In  order  to  find 
the  length  of  a  pendulum  that  will  oscillate  a  certain  num- 
ber of  times  in  a  minute,  make  this  proportion :  *'  t  as  the 
square  of  the  given  number  of  oscillations  is  to  the  square 
of  60,  or  the  number  of  seconds  in  a  minute,  so  let  the 
standard  pendulum,  or  39'128  inches,  be  to  the  pendulum 
sought. 

Example. — Required  the  length  of  a  pendulum  which 
will  vibrate  20  times  per  minute. 

*  See  Young's  Natural  Philosophy,  vol.  i.  p.  47. 
t  Anderson's  Institutes  of  Physics,  vol.  i.  p.  250. 


AT  T.3                         MOTION    OP   MILLS. 

31 

Feet.     Inches.    Feet 
0x"o:'2S*"^3liOO::3..S-l.8:«. 
lengtli  required. 

Inehe. 

.-i-ia 

SECTION  I. 

L  395,  In  a  windmill,  when  the  velocity  is  increased  by 
!  irregular  action  of  the  wind,  the  com  is  sometimes 

arced  rapidly  through  the  mill  without  being  sufficiently 
ground.  There  is  an  elegant  contrivance  for  preventing 
this,  (similar  to  the  governor  of  a  steam-engine,)  but  which 
I  believe  was  much  earlier  xn  use,  called  in  some  parts  of 
England  a  Lift-  Tenter.  "  By  means  of  the  centrifugal 
force  of  one  or  more  balls,  which  6y  out  as  soon  as  the  ve- 
locity is  augmented,  and  as  the  rise  in  the  arc  of  a  circle, 
allow  the  end  of  a  lever  to  rise  with  them,  while  the  oppo- 
site end  descends  witli  the  upper  millstone,  and  brings  it  a 
little  nearer  to  the  lower  one."  * 

This  machine  is  curious,  and  might  perhaps  in  other 

ises  be  usefully  applied.     We  shall,  therefore,  describe 
3  constructions,  but  both  on  the  same  principles. 


L1FT-TENTBB3  FOR  WINIIHILLS. 


First  Conttruetim, 


196.  This  machine  and  part  of  the  stone-spindle  and 

ning  with  which  it  is  connected,  are  represented  in  Fig. 

.  Plate  XIII. 

To  the  stone-spindle  there  are  fixed  four  arms  a,  a,  a,  a, 

there  are  four  similar  arms  b,  b,  b,  b  firmly  attached  to  the 

hollow  cylinder  c,  which  is  loose  on  the  spindle  fg. 

The  pendulums  d,  d,  d,  d  are  hung  above  to  the  arms 

*  Young's  Natumi  PliiJi)Bo|ib_v.  vol.  i.  p.  233. 


SIS  ON   EQUALIZING   THE  [eSSATT. 

A,  A,  A,  Ay  and  through  holes  toward  their  lower  extremi- 
ties pass  the  arms  of  the  loose  cylinder. 

When  the  mill  is  at  rest,  the  pendulums  hang  vertically ; 
hut,  hy  their  centrifugal  force,  when  the  miU  is  in  motion 
they  hang  ohliquely ;  and  that  ohliquity  is  increased  in 
proportion  to  the  velocity,  and  proportionately  raises  the 
loose  cylinder  c. 

This  cylinder  c  acts  on  the  one  end  of  the  lever  e,  which 
has  a  connexion  with  the  clove  upon  which  the  bridge  of 
the  stone-spindle  rests,  and  accordingly  raises  or  depresses 
the  upper  millstone  in  proportion  as  the  wind  is  weak  or 
strong. 


Second  CatuintetiaH, 

397.  Another  modification  of  the  same  principle,  (ap- 
plied  above  the  millstones ,)  but  having  one  pendulum 
only,  is  represented  by  Fig.  4,  and  will  be  easily  under- 
stood  from  what  has  been  said  respecting  the  First  Con- 
struction. 

These  lift-tenters  are  drawn  from  sketches  taken  in  the 
neighbourhood  of  Liverpool  in  the  year  1790. 


SECTION  III. 

398.  Governors  are  sometimes  applied  to  water-wheels, 
and  made  on  various  constructions.  Smiths*  bellows  have 
been  applied  to  that  use,  the  upper  board  rising  or  falling 
in  proportion  to  the  velocity  of  the  lower  board,  which  re- 
ceived its  motion  from  the  mill.  But  those  we  are  about 
to  describe,  appear  to  me  on  better  principles ;  and  as  they 
have  been  found  of  very  material  use,  we  shall  proceed  to 
describe  a  construction  which  has  for  several  years  been  at 
work  in  Cartside  cotton-mill,  which  was  erected  under  the 
direction  of  the  late  Robert  Bums,  Esq.,  (of  whom  Pro- 


ESSAT  v.] 


MOTION    OP    MILLS. 


313 


WATER-WHEEL  GOVBBNOB. 


First  Comtruetion 


fessor  Robinson  makes  respectful  mention  in  the  Encyclo- 
psedia  Britannica,  Art.  Water-works.)  and  which  has  there 
A«en  gireat  satisfaction  *  ;  we  shall  afterward  describe  some 
H^r  similar  machines  for  the  same  use. 

^899.  The  principles  of  this  kind  of  water-wheel  governor 
are  nearly  the  same  as  those  of  the  governor  of  a  steam 
engine.  It  has  a  revolving  pendulum  which  receives  its 
motion  from  the  mill,  and  in  proportion  as  the  machinery 
moves  faster  or  slower,  the  centrifugal  force  acts  upon  the 
governor,  and  raises  or  depresses  an  iron  cross,  which, 

Eg  on  a  lever,  reverses  the  motion  by  the  wheel-work, 
h  operates  upon  a  sluice  so  as  to  enlarge  or  lessen  the 
ige  of  the  water  to  the  water-wheel ;  this  sluice  is 
made  on  the  principles  of  the  ihrotUe-valve  already  de- 
scribed, Art.  388,  in  order  that  it  may  be  moved  by  a 
Htall  power.  So  long  as  the  machinery  is  moving  at  a 
Beper  velocity,  this  wheel-work  of  the  sluice  apparatus 
remains  at  rest. 

Fig.  5  represents  different  views  of  this  machine,  and 
some  of  its  parts  detached.  The  same  letter  in  all  the 
figures  refers  to  the  same  part. 

The  revolving  pendulum  efgh  receives  its  motion  from 
the  mill-work  by  means  of  a  rope  giving  motion  to  a  pulley 
I.     The  upright  shaft  mn  is  kept  in  constant  motion  by 

n  a  letter  which  Buchanan  hod  from  Mr.  Bums,  dated  February  1B08, 
TJtes  to  tlie  following  effect ; 

rhe  goTemor  is  the  most  useful  thing  For  a  wftter-whecl  that  can  pos- 

V  be  thought  of,  and  I  wish  you  would  adopt  it  in  your  practice  wherever 

{doMrs  to  make  your  employer  prosper.     1  am  sure  it  ia  worth  a  lai^ 

<  us  at  CarlAidc  mill,  from  its  keeping  up  the  speed  of  the  machinery, 

it  de'viUJng  the  yeitr  round." 


314  ON   EQUALIZING  THE  [E881T  Y. 

the  wheel  work  oprb.  The  wheel  n  acts  constantly  into 
the  two  hevelled  wheels  t  and  u,  and  makes  them  move  in 
contrary  directions.  They  are  loose  on  the  shaft  when  the 
miU  is  going  at  its  proper  speed. 

But  if  the  mill  moves  either  too  £Btst  or  too  slow,  the  one 
of  these  wheels,  hy  means  of  a  clutch  q,  in  a  way  to  be 
described,  is  connected  with  and  carries  round  the  lying 
shaft  D  c,  and,  by  a  pair  of  bevelled  wheels,  communicates 
motion  to  the  oblique  shaft  b  w,  which  again,  by  a  screw  x, 
and  quadrant  wheel  t,  moves  the  sluice  z,  and  by  making 
it  stand  more  or  less  oblique,  alters  the  area  of  the  passage 
for  the  water. 

From  inspecting  Fig.  5,  No.  1,  it  will  be  evident  that 
the  box  a  will  be  raised  or  depressed  in  proportion  as  the 
baUs  E  and  f  of  the  revolving  pendulum  efgh  are  further 
or  nearer  to  the  centre  of  motion ;  when  the  velocity  is 
greatest,  the  balls  e  and  f  by  their  centriftigal  force  will 
extend  themselves  farthest  from  the  centre  of  motion,  and 
raise  the  box  a.  See  also  Iig.  5,  No.  S,  No.  3,  and 
No.  4. 

To  the  box  a  is  fixed  a  cross  be.  There  is  a  forked 
lever  dqe^  the  fulcrum  of  which  is  at^  and  which  turns 
horizontally.  This  forked  lever  has  four  prongs,  1,  2, 
3,  4. 

^V^lcn  the  mill  is  at  its  proper  speed,  the  cross  works  within 
the  prongs  1  and  2 ;  in  this  situation  of  the  forked  lever 
the  clutch  Q  is  disengaged  from  both  the  wheels  t  and  u, 
and  they  move  on  their  bushes  without  carrying  round  the 
lying  shaft.  The  clutch  is  made  to  slide  on  a  part  of  the 
shaft  which  is  square. 

When  the  mill  goes  too  quick  the  cross  gland  is  raised, 
and  in  turning  round  strikes  the  prong  3,  which  immedi- 
atelv  causes  the  lever  to  throw  the  clutch  into  the  arms  of 
the  wheel  u,  which  then  carries  the  clutch  and  shaft  round 
with  it,  and  by  the  means  already  described  acts  on  the 


E88AT  y.3  MOTION   OF  MILLS.  did 

shiice,  and  by  lessening  the  quantity  of  water  falling  on 
the  wheels  diminishes  its  speed. 

On  the  other  hand,  when  the  mill  goes  too  slow,  the 
cross  is  depressed,  and  striking  the  prong  4,  reverses  the 
motion  of  the  shaft,  and  so  produces  a  contrary  e£Fect  on 
the  sluice. 

400.  It  may  be  proper  to  remark,  that  the  train  of  wheel- 
work  is  so  calculated  as  very  much  to  reduce  the  motion  at 
the  sluice,  and  it  is  found  firom  experience  that  this  is  ne- 
oetrary.  Were  the  area  of  the  aperture  ttio  suddenly 
changed,  the  effect  on  the  water*wheel  would  be  too  vio- 
lent. Every  time  the  mill  is  stopped,  it  is  proper  to  lift 
the  wheel  r  out  of  gear.  The  centre  on  which  the  sluice 
turns  should  be  one  third  of  its  height  firom  the  bottom,  in 
order  that  the  pressure  of  the  water  above  the  centre  may 
balance  that  below. 

At  m  there  is  an  upright  shaft,  which  is  worked  by  hand 
when  required. 

WATBR-WHBBL  OOVBBNOB. 

Second  Conttmctum. 

401.  Fig.  6  represents  a  sluice  regulator  as  executed  in 
some  parts  of  England.  It  differs  little  fi:'om  that  already 
described,  only  that  the  lying  shaft  a  b  receives  its  motion 
immediately  firom  the  mill,  instead  of  firom  the  axle  of  the 
revolving  pendulum,  as  in  the  first  construction.  (Art.  399*) 
From  having  so  minutely  described  that  construction,  it  is 
hoped  that  the  attentive  reader  will  find  no  difficulty  in 
ccmiprehending  Fig.  6,  firom  inspecting  the  plate. 

WATBB-WHSBL  GOVBRNOB. 

Third  Construction. 

ifiSL  Fig.  7»  Plate  XV.  represents  a  water-wheel  go- 
vernor of  a  very  simple.  cpziBtractiaQ»  differing  £rom  the 


316  ON  EQUALIZING  THE  [e88AT  V. 

foregoing  in  this  respect,  that  it  communicates  most  part 
of  its  motion  by  bands  and  pulleys  instead  of  wheel-work. 
The  motion  is  reversed  by  the  simple  means  of  having  one 
of  the  pulleys  a  with  an  open  band,  and  the  other  b  with 
a  cross  band. 


OBSERVATION. 


It  is  proper  to  observe  here,  however,  as  was  already 
done  with  regard  to  the  governor  of  the  steam  engine,  that 
wheel- work  is  much  to  be  preferred  in  point  of  certainty, 
to  bands  and  pulleys. 


WATBB-¥rHBSL  OOVBBNOIU 

Fourth  Constmetion, 

403,  This  construction  is  represented  in  Fig.  8.  The 
revolving  pendulum  aklm  is  kept  in  constant  motion  by 
the  water  wheel. 

a,  a,  two  wheels  fixed  on  round  sockets  upon  the  go- 
vernor spindle. 

B,  a  clutch  upon  a  square  part  of  the  spindle,  (or  what 
might  be  better,  a  round  with  a  feather  upon  one  side ;)  c,  a 
gland  to  connect  the  clutch  with  the  sliding  part  k  of  the 
governor,  which  has  a  groove  to  receive  it  like  that  of  a  bay* 
onet.  (See  Essay  IV.  Art.  282.)  d,  a  piece  of  iron  which 
prevents  the  gland  from  turning  round,  and  for  keeping  it 
from  flying  off^;  e,  a  wheel  working  into  the  wheels  a,  a  ; 

F,  an  endless  screw  upon  the  same  axle  with  the  wheel  e  ; 

G,  a  wheel  upon  the  same  axle  with  another  screw  h,  which 
acts  into  the  quadrant  i,  upon  the  sluice. 

The  operation  of  this  ingenious  apparatus,  from  what 
has  been  said  of  the  other  constructions,  will,  it  is  hoped, 
be  sufficiently  clear.  This  governor  was  designed  by  Jam^ 
Carmichael,  millwright,  of  Dundee. 


ESSAY  V.3  MOTION    OF   MILLS.  317 


WATBB-WHEBL  OOVBRNOB. 

Fifth  Camtructian, 

404.  From  the  inspection  of  Fig.  9^  and  what  has  been 
ahready  said  respecting  water-wheel  governors,  this  par- 
ticular construction  will  doubtless  be  easily  comprehended. 
We  need  only  mention  that  the  wiper  a,  by  means  of  the 
forked  lever  bdc,  acts  on  the  clutch  e.  The  rest  of  the 
movements  resemble  those  of  the  first  construction.    (Art. 

3990 

This  apparatus,  remarkable  for  its  neatness  and  simpli- 
city, was  constructed  by  Hewes  of  Manchester*. 

*  A  simple  and  not  a  very  expensive  apparatus  for  equalizing  the  exer- 
tion of  horses  in  thrashing-machines,  is  descrihed  in  the  Art.  Agriculture, 
Supplement  to  Ency.  Brit  p.  200 ;  and  in  Brewster's  edition  of  Feiguson's 
Lectoies,  p.  201,  vol.  ii. 


1.  <i 


APPENDIX. 


ON  THE 


VELOCITY    OF    WATER-WHEELS. 


No.  I. 

On  the  Velocity  of  Water- Wheels^  by  Mr.  Robertson 
Buchanan^  Engineer :  communicated  in  a  Letter  ta  the 
Editor  of  the  Philosophical  Magazine. 

The  accompanying  paper  was  read  in  May  1799>  ^  * 
Philosophical  Society  at  Edinhurgh,  and  was  afterwards 
published  in  the  Philosophical  Magazine. 

405.  There  are  many  cases  in  which  it  is  of  importance 
to  know  the  proportion  of  power  necessary  to  give  different 
degrees  of  velocity  to  a  mill*.  But  as  the  construction  of 
mills,  and  the  purposes  they  serve  are  various,  it  is  perhaps 
impossible  to  find  any  law  of  universal  application.  Mr. 
Banks,  in  his  Treatise  on  Mills  i,  has  drawn  a  conclusion 
which  he  appears  to  consider  as  invariable,  namely,  that 
"  when  a  wheel  acts  by  gravity,  its  velocity  will  be  as  the 
cube  root  of  the  quantity  of  water  it  receives." 

But  if  we  suppose  a  wheel  raising  water  by  means  of 
cranks  and  pumps,  on  Mr.  Bank^s's  principle,  Buchanan 

*  It  was  a  scarcity  of  water  for  the  Rothesay  mills  which  directed  my 
attention  particularly  to  this  suhject 

t  See  Banks  on  Mills,  pp.  17,  18,  144,  145,  146. 


APPEND.]       ON    THE    VELOCITY    OF    WATER-WHEELS.  319 

thought  it  might  easily  be  demonstrated,  that  by  reducing 
the  velocity  of  the  wheel  to  a  certain  degree,  the  wheel 
would  raise  more  water  than  would  be  necessary  to  move  it 
at  that  velocity  ;  a  thing  evidently  impossible. 

In  this  view  it  would  seem  there  is  no  actual  case  in 
which  Mr.  Banks's  conclusions  will  hold  true.  But,  how- 
ever they  may  apply  to  other  mills,  the  experiments  of  Bu- 
chanan seem  to  prove  at  least  that  they  do  not  apply  to 
cotton-mills.  On  the  ground  of  these  experiments,  made 
at  different  times,  and  with  all  the  attention  in  our  author's 
power,  (and  not  from  any  abstract  consideration,)  did  he 
presume  to  call  in  question  an  authority  for  which  we  en- 
tertained the  highest  respect. 

406.  In  January  1796  he  measured  the  quantity  of 
water  the  Rothesay  old  cotton-mill  required :  1st.  When 
going  at  its  common  velocity ;  and  2dly,  when  going  at 
half  that  velocity.  The  result  was,  that  the  last  required 
just  half  the  quantity  of  water  which  the  first  did.  It  is 
to  be  observed,  that  in  these  experiments  the  quantities  of 
water  were  calcidated  from  the  heads  of  water  and  aper- 
tures of  the  sluices. 

feFrom  these  experiments  he  inferred,  "  That  the  quantity 
water  necessary  to  be  employed  in  giving  different  de- 
Bes  of  velocity  to  a  cotton-mill,  must  be  nearly  as  that 
velocity." 

He  was  satisfied  with  this  experiment,  and  the  inference 
drawn  from  it,  till  some  gentlemen  well  acquainted  with  the 
llieorj-  and  practice  of  mechanics  expressed  their  doubts  on 
the  subject.  He  had  then  recourse  to  another  experiment, 
which  he  considered  as  less  liable  to  error  than  the 
former. 

■toy.  The  water  which  drives  the  old  cotton-mill  falls,  a 
little  below  it,  into  a  perpendicular-sided  pond,  which 
serves  as  a  dam  for  a  corn-mill  at  some  distance  below  it. 
To  ascertain,    therefore,    the    projiortional    quantities    of 

y2 


320 


ON  THE   VELOCITY 


[essay  Y. 


water  used  by  the  old  mill,  notmiig  more  was  necessary 
than  to  measure  the  time  the  water  took  to  rise  to  a  eertahi 
height  in  that  pond ;  and  accordingly,  on  the  first  of  May 
1798,  he  made  the  experiments  noted  in  the  following 
table : 


Number  of  experiments. 

• 

1 

2 

3 

4 

Revolutions  of  one  of  the  upright 
shafts  per  minute. 

46 

46 

24 

23 

Rise  of  water  in  the  pond  in  inches. 

5 

5 

5 

5 

Time  in  minutes  and  seconds. 

6-58 

6-57 

14-45 

150 

The  first  and  second  experiments  were  made  with  the 
mill  at  its  common  velocity ;  the  third  and  fourth  at  nearly 
half  that  velocity. 

The  time  which  the  mill  required  to  use  the  same  quan- 
tity of  water  in  these  experiments  may  be  taken  in  round 
numbers ;  the  proper  velocity  at  7  minutes,  and  half  that 
velocity  at  15  minutes. 

The  result  of  these  experiments  approaches  very  nearly 
to  that  of  1796.  The  difference  may  be  accounted  for  by 
the  small  degree  of  leakage  which  must  have  taken  place 
at  the  sluices  on  the  lower  end  of  the  pond ;  and  the  time 
being  greater  in  the  third  and  fourth  experiments,  the 
leakage  would  of  course  be  greater. 

408.  Smeaton  *  and  others  have  proved,  in  a  very  satis- 
factory manner,  that  "  the  mechanic  power,  that  must  of 
necessity  be  employed  in  giving  different  degrees  of  velo- 
city to  the  same  body,  must  be  as  the  square  of  that  velo- 
city." But  it  appeared  to  Buchanan,  that  the  result  of 
the  above  experiments  may  be  easily  reconciled  to  this  pro- 
position,  by  considering  what  Smeaton  says  immediately 

*  Sec  Smeaton  on  Mills,  p.  18.     See  his  Miscellaneous  Papers,  p.  92. 


^PEND.] 


OF    WATER-WHEELS. 


321 


terwards: — "If  the  converse  of  this  proposition   (says 

)  did  not  hold  true,  viz.,  that  if  a  body  in  motion,  in 
eing  stopped,  would  not  produce  a  mechanical  effect  equal 
or  proportional  to  the  square  of  its  velocity,  or  to  the  me- 
chanical power  employed  in  producing  it,  the  effect  would 
not  correspond  with  its  producing  cause.""  Now  it  is  to 
be  observed,  that  Smeaton's  experiments  were  made  on  the 
velocihi  of  heavy  bodies  Jree  from  frictioyi  and  other  causes 
nf  resistance. ;  but  in  mills  there  is  not  only  friction,  but 
obstacles  to  ho  removed  :  and  experiments  made  on  friction 
have  proved  that  the  frictions  of  many  kinds  of  bodies  in- 
crease in"  direct  proportion  to  their  velocity.  But  the  velo- 
city of  a  cotton-mill  at  work  may  be  considered  as  a  me- 
chanical effect;  and,  if  so,  must  correspond  with  its  pro- 
ducing cause. 

409.  The  preceding  esperiments  on  the  Rothesay  mill, 
are  undoubtedly  correct  and  consistent  with  the  principles 
of  motion  and  power,  and  also  with  the  experiments  of 
Smeaton  on  Mills  and  Mechanic  Power. 

It  is  shewn  in  the  additions  to  this  essay  that  the  me- 
chanical power  is  as  the  quantity  of  water  on  the  wheel, 
multiplied  into  its  velocity  when  the  wheel,  fall,  and  other 
circumstances  remain  the  same,  and  since  the  mechanical 
effect  is  measured  by  the  resistance  multiphed  into  the 
velocity  of  the  working  point  when  the  friction  is  con- 
stant ;  if  the  quantity  of  water  be  diminished  by  its  half, 
either  half  the  resistance,  or  half  the  velocity  with  which 

is   overcome,    must   be    taken    away,    otherwise    there 

I  not  be  an  equilibrium  between  the  power  and  effect. 

at    the   same    time   it   is   to  be   observed,    that    an 

•eased  velocity  lessens  the  friction  of  the  intermediate 

ichinerj',  and  consequently  a  greater  effect  would  be  pro- 
Ked  by  the  greater  velocity,  as  appears  to  be  the  case  by 
1  Mechanie  Powers  applied  to  Bodies  at  rest.    Miscellaneous 


3^  ON    TH£   VELOCITY  [eSSAT  V. 

the  experiments*  There  is  not,  however,  in  the  detail  of 
these  experiments,  sufficient  data  to  enahle  ub  to  arrire  at 
any  useful  conclusions. 

410.  Roherton,  an  engineer  of  some  eminence,  made 
ohservations  on  the  foregoing  paper,  alleging  that  the 
conclusions  of  Banks  give  most  satisfactory  evidence 
that  particular  care  and  judgment  are  necessary  in  "*ffVrf 
experiments. 

It  appeared  to  Roherton  that  the  wrong  oonchuoanB 
which  have  heen  drawn  hy  this  and  other  writers  on  ihiB 
suhject  have  wholly  arisen  from  misapprehending  some  of 
Sir  Isaac  Newton's  fundamental  principles  of  mechanici^ 
and  from  a  love  of  establishing  theoretical  expressiape 
rather  than  strict  observations  of  the  invariable  laws  of  na- 
ture ;  expressions  such  as  these :  viz.,  Quantity  qfUdotkn^ 
Instantaneous  Impulse. 

Taking  a  constant  portion  of  time  (viz.,  a  second)  to  be 
the  measure  of  the  velocity  of  a  body,  and  an  instant  to  be 
the  measure  of  the  effect  it  produces ;  or  by  taking  time 
as  the  measure  of  the  cause,  and  space  as  the  measure  of 
the  effect.  As  to  an  instantaneou.s  effect,  Roherton  ar- 
gues that  it  is  an  absurdity  in  itself  as  well  as  in  mechanics. 
We  can  form  no  idea  of  a  body  put  into  motion,  without 
the  acting  power  or  body  act  upon  the  body  put  into  motion 
for  some  timey  and  also  over  some  space ;  and  to  suppose 
otherwise  leads  us  entirely  out  of  the  sound  principles  <tf 
mechanics. 

In  mechanics  every  effect  is  equal  to  its  producing  cause. 
In  the  case  of  a  power  acting  on  a  body  producing  motian» 
and  also  this  body  acting  against  another  power  which 
tards  its  motion :  if  the  causes  of  action  and  resistance 
each  measured  by  the  tbne  the  motions  are  produced  and 
retarded,  the  result  will  be  equal. 

Or  if  they  be  measured  by  the  space  over  which  tbey 
act,  the  results  will  be  equal ;  and  this  is  an  uid¥B(aa]l 


I  APPEND.]  OF    WATER-WHEELS.  SS3 

principle,  whether  applied  to  accelerating  power  and  mo- 
tion, as  gravity,  &c.,  or  to  machines  which  act  constantly 
and  uniformly.     Yet  in  the  case  of  uniform  motion,  space 
or  time  may  be  used  at  pleasure ;  as  from  the  uniformity 
of  space  and  time  they  become  a  common  measure- 
To  illustrate  this,  suppose  the  body  a  acted  upon  by  the 
power  of  gravity  through  the  space  ab,   in  a  portion  of 
time  which  we  will  call  one,     M'hen  it  arrives  at  b,     ^  q 
it  meets  with  another  medium  of  resistance,  which 
is  ten  times  greater  than  the  former ;  the  body  a 
will  be  resisted  in  proportion  to  the  cause  of  action 
and  resistance,  that  is  to  say,  if  the  time  of  action 
were  one  second,  the  tijne  of  resistance  will  be  one 
tenth  of  a  second,  and  the  distance  ab  will  be  to 
the  distance  bc    as  ten  to  one;   so  that  whether 
space  or  time  be  taken  as  the  measure  of  action, 
the  same  must  be  taken  for  the  measure  of  the 
eflfect,  to  have  the  results  proportionate  and  equal. 
But  if  the  cause  be  measured  by  time,  and  the  effect 
by  space,  the  results  will  be  as  the  squares  of  the  times,  or, 
which  is  the  same  thing,  as  the  squares  of  the  velocity. 
^^    Thus,  suppose  a  body  in  motion,  with  a  velocity  of  one, 
^Hm  a  power  to  penetrate  into  a  hank  of  earth  one  foot. 
^Ki  the  same  body,  with  a  velocity  of  two,  strike  the  bank, 
It  will  penetrate  to  the  depth  of  four  feet:  for  the  velocity 
is  double,  and  the  time  of  action  is  double,  and  therefore 
the  results  will  be  compounded  of  both,  that  is  as   the 
square  of  the  velocity. 

From  the  above  it  may  be  inferred,  that  if  equal  bodies 
be  acted  upon  by  unequal  powers,  the  times  requisite  to 
produce  an  equal  motion  will  be  reciprocally  proportionate 
to  the  powers  ;  that  is  to  say,  if  a  power  of  ten  act  upon  a 
body  for  one  second  of  time,  and  the  power  of  one  act  upon 
an  equal  for  ten  seconds,  thev  will  produce  equal  velocities. 


1 


324  ON   THE   VELOCITT  [M84T.T. 

very  unequal,  being  as  ten  to  one :  and  if  ihe  squan  nMb 
of  the  powers  producing  the  effects  be  taken,  that  will  give 
the  times  they  take  in  carrying  the  body  acted  upon  through 
equal  spaces. 

But  it  is  obvious  this  doctrine  has  no  more  to  do  wifli 
the  operation  of  machines,  than  simply  their  first  startnig 
from  rest  to  the  motion  necessary  for  working.  When 
this  is  acquired,  the  power  applied  and  the  power  of  resiit- 
ance  balance  each  other,  and  whatever  be  the  motioii  the 
machine  moves  at,  the  same  power  will  carry  it  on,  (if  it  he 
upheld,)  provided  the  machine  act  in  such  a  manner  as  not 
to  accumulate  resistance  by  the  accumulation  of  motijon, 
which  is  the  case  in  forcing  fluids  through  pipes,  &c.  Li 
cases  of  this  kind,  the  nature  of  the  machine  must  be  par- 
ticularly kept  in  view,  and  not  to  adopt  any  law  to  explain 
the  resistance  the  acting  body  meets  with,  but  what  is  am* 
ply  deduced  from  the  very  machine  we  have  under  cod* 
sideration.  But  in  most  cases,  any  machine  may  be  cod* 
sidered  as  acting  purely  on  a  statical  principle.  The  rais- 
ing of  weights,  or  overcoming  friction,  Roberton  considerB 
purely  as  acting  on  that  principle ;  and  when  the  power  of 
action  is  equal  to  the  resisting  power,  the  machine  is  indif- 
ferent to  motion  or  rest.  If  the  machine  be  at  rest,  the 
power  will  not  move  it,  being  a  balance  to  the  resistance. 
If  the  machine  be  set  in  motion,  the  power  will  keep  it  in 
the  same  motion,  (provided  the  power  be  upheld,)  the 
same  as  equal  weights  himg  over  a  pulley,  or  in  the  oppo- 
site scales  of  a  beam.  If  they  be  at  rest,  they  will  remain 
so ;  and  if  they  be  put  in  motion,  they  will  endeavour  to 
persevere  in  the  same. 

The  above  doctrine  of  a  statical  principle  is  proved  in 
the  most  satisfactory  manner  by  the  experiments  made  at 
the  old  mill  of  Rothesay,  the  motion  of  the  water-wheel 
being  exactly  proportioned  to  the  quantity  of  water  ez» 
pended,  and  therefore  an  exact  and  equal  load  upoii  •  the 


APPEND.]  OP    WATER-WHEELS.  325 

wheel ;  that  is  to  say,  the  buckets  icere  equally  full  when 
the  mill  moved  at  its  ordinary  motion^  or  at  half  that  mo- 
tion. 

The  effect,  therefore,  of  letting  more  water  on  a  wheel, 
is  not  to  lodge  a  greater  quantity  in  the  buckets,  but  to 
supply  the  same  quantity  when  the  wheel  is  in  a  greater 
motion. 

Banks,  however,  made  his  experiments  agree  with  his 
theory,  yet  Roberton  took  no  trouble  in  enquiring  into 
them^   alleging  it  would  be  to  little  purpose  to  have  done 


»r 


Bus 


Suffice  it  to  say,"  he  adds,  "  that  the  very  small  quan- 
bties  of  water  which  Banks  made  use  of,  and  the  slowness 
of  the  motion  of  his  wheel  in  his  experiments,  gives  no 
ground  for  placing  the  smallest  dependence  on  them,  and 
when  compared  with  the  more  judicious  and  accurate  ex- 
periments of  Smeaton,  they  dwindle  into  contempt." 

411.  Roberton  further  says  that  "Smeaton,  in  running 
wheel  at  nearly  three  feet  in  the  second,  brought  it 
nearly  to  a  maximum,  and  lost  but  about  one  fourth  or 
one  fifth  of  the  original  effect  (alludiug  to  his  overshot 
wheels).  Banks,  at  his  highest  motion,  run  his  wheel 
about  one  foot  in  the  second,  and  reducing  it  to  one 
half  of  that  motion,  the  same  quantity  of  water  then  ex- 
pended, was  capable  of  performing  four  times  the  wurk  ; 
and  by  deduction  from  thence,  it  appears  plain  that  his 
wheel,  (from  his  own  theorj',)  would  perform  about  twenty 
times  the  quantity  of  work  which  iSmeaton's  could  per- 
form with  the  same  quantity  of  water,  and  about  16 
more  than  nature ;  so  that  the  observation,  (alluding 
the  theory  of  Banks,)  is  very  just  in  sajing  that,  by  re- 
\g  the  motion  of  the  wheel,  it  is  demonstrable  it  would 
more  water  than  supply  itself." 


326 


ON  THE  VELOCITY 


[essay  V. 


ON   0VBB8H0T   WATBA-WHBKL8. 


412.  The  best  water-wheel  is  that  which  is  calculated  to 
produce  the  greatest  effect  when  it  is  supplied  bj  a  stream, 
furnishing  a  given  quantity  of  water,  with  a  given  &IL 

The  mechanical  effect  depends  on  the  proportion  of  the 
wheel's  diameter  to  the  height  of  the  fall ;  and  on  the  ve- 
locity of  the  circumference  of  the  wheeL  These  are  the 
two  principal  parts  to  be  considered  in  the  theory  of  wheels ; 
but  there  are  also  some  other  points  which  ought  to  be  at- 
tended to,  because  the  effect  is  much  decreased  when  they 
are  neglected. 

Of  the  proportion  of  the  radius  of  the  wheel  to  the 
height  of  the  fall. — ^Let  acbd  be  the  wheel,  and  ea  the 


depth  of  the  buckets ;  then,  according  to  experiments  on 
water-wheels,  it  appears  that  the  rotary  force  of  the  water 
in  the  buckets  is  nothing  at  c  and  d,  and  that  it  increases 
nearly,  if  not  accurately,  in  the  direct  ratio  of  the  distance 


OF    WATER-WHEELS. 

from  c  or  rf,  and  is  greatest  at  a.  That  is,  the  force  at ' 
any  poiot  n  in  a  direction  ea,  or '  perpendicular  to  the 
radius,  is  as  ac. 

A  slight  consideration  of  the  figure  is  sufficient  to  in- 
form us,  that  the  wheel  will  not  produce  the  greatest  effect 
when  it  receives  the  water  at  the  upper  pomt  c,  and  that 
there  must  be  considerable  advantage  in  making  the  wheel 
of  a  greater  diameter,  so  that  it  may  receive  the  water  at 
some  point  between  a  and  c.  The  point  which  will  ensure 
the  greatest  efiect  we  are  now  to  calculate. 

Put  c  =  that  portion  of  the  circumference  which  is  to  he 

loaded  with  water ;  and  .r  =  the  arc  comprehended  between 

the  point  where  the  water  flows  upon  the  wheel  and  the 

horizontal  line  ea  ;  also  make  b  =  the  area  of  the  stream 

supplying  the  buckets.     Then  the  solid  which  represents 

<?  —  '^ 

the  effective  force,  will  be  4  5  x  I ^  ;  which  is  to  be 

'^  c—x  ' 

c*  — 2j* 
the  greatest  possible ;  or  — ^^—  =  a  maximum.    By  the 

principles  of  maxima  and  minima,  this  takes  place  when 
j-=c(l—  ■j\)OT  X  =  ''29'29  c.  Accordingly  the  arc  c  —  x 
must  be  the  quadrant  dg  or  90",  and  the  arc  x  =  37'27°. 

Hence  we  have  this  important  practical  maxim,  A 
water-wheel  will  produce  the  greatest  effect  when  the  dia- 
meter of  tlie  wheel  is  proportioned  to  the  height  of  the  fall, 
so  that  the  water  flows  upon  the  wheel  at  a  point  about 
52j  degrees  distant  from  the  summit  of  the  wheeL 

If  r  be  the  radius  of  the  wheel  to  the  extreme  part  of 
tfie  bucket,  and  h  the  effective  height  of  the  fall,  then  k  =  r 
w(l+sin.  87^,)  or  A  =  1-605  r;  for  the  sin.  37i  =  -605. 
E|&lso  'GSS  h~T.  Therefore  when  the  effective  height  of 
Kdie  fall  is  determined,  the  radius  of  the  wheel  is  easily 
^lnJculated.  When  the  effective  fall  is  §  of  the  whole  fall, 
^  if  we  make  h  the  whole  fall,  r  =  ■554'/i,  or  I'lOS  A  =  the 
diameter  of  the  wheel. 


328  ON   THE   VELOCITY  [eSSAT  V. 

The  effective  height  of  the  fall  is  less  than  the  true 
height,  hy  as  much  as  is  necessary  for  giving  the  water  the 
same  velocity  as  the  wheel  hefore  it  flows  upon  it. 

In  low  falls  a  wheel  would  work  with  advantage  in  a 
considerable  depth  of  tail  water,  provided  the  buckets  were 
of  a  suitable  form  for  moving  through  the  water,  and  the 
effective  fall  made  through  a  very  accurate  sweep,  so  that 
the  sweep  and  not  the  form  of  the  bucket  should  confine 
the  water  upon  the  wheel. 

413.  Of  the  velocity  of  the  circumference  of  the  whed 
to  produce  a  fnasimum  effect. — It  is  necessary  to  premise, 
that  the  velocity  with  which  the  water  flows  upon  the  float- 
boards  or  buckets,  is  considered  to  be  equal  to  the  velocity 
of  the  wheel,  and  to  strike  against  the  floats  as  nearly  as 
possible  in  the  direction  of  the  motion  of  the  wheeL 

Let  X  be  that  part  of  the  fall  which  gives  the  necessary 
velocity  v  to  the  water,  when  the  effect  is  a  maximum ; 

V  will  then  be  the  velocity  of  the  circumference  of  the 
wheel.  Also,  make  a  =  that  part  of  the  fall  which  would 
correspond  to  the  velocity  of  the  circumference  of  the 
wheel  when  the  power  would  be  equal  to  the  friction  of 
the  loaded  machine  only ;  or  when  the  useful  effect  would 
be  nothing.  Now  if  A  be  the  whole  fall,  the  effective  force 
of  the  water  on  the  wheel  will  always  be  proportional  to 
A  —  jT,  when  the  effect  is  a  maximum ;  and  to  A  —  a,  when 
the  useful  effect,  or  work  done,  is  nothing. 

Hence,  v  (^h  ^  x  —  h  —  a)  must  be  a  maximum ;  or, 

V  (a  —  ^)  =  a  max.,  but  v  =  x^,  therefore  jr*  (a  ^  jr)  =  a 
max.,  which  according  to  the  rules  of  maxima  and  tninima^ 
takes  place  when  a  =  3x. 

It  is  evident  that  the  value  of  a  must  entirely  depend  on 
the  nature  of  the  machine,  for  if  there  be  many  moving 
parts  between  the  power  and  the  resistance,  the  firiction 
will  be  greater,  and  consequently  a  will  be  less.  The  m^ 
chine  must  be  very  simple  indeed,  if  the  friction  be  less 


JPEND.]  OP    WATER-WHEELS.  329 

I  one  half  the  moving  power,  and  it  will  often  amount 
I  thirds  of  it.      If  we  suppose  it  to  be  two  thirds, 

len  a  =  o,  and  consequently  x  =  t.,  and  u  =  V  — j^ —  = 

^  h. 

Hence,  when  the  friction  amounts  to  two  thirds  of  the 
moving  power,  the  velocity  of  the  circumference  of  an  over- 
shot wheel  in  feet  per  second,  should  be  2*67  times  the 
square  root  of  the  whole  height  of  the  fall  in  feet. 

Again,  that  part  of  the  fall  is  to  be  determined,  which 

will  give  the  water  the  same  velocity  as  the  wheel,  and 

h  h 

=  5,  and  3jr  =  a,  we  have  ^  =  fi-    Hence,  when  the 


^Bnce 

^picti' 


iction  is  two  thirds  of  the  power,  that  part  of  the  fall 
which  will  give  the  water  the  proper  velocity,  is  one  ninth 
of  the  whole  height, 

These  results  may  now  be  usefully  compared  with  the 
experiments  of  Smeaton  ;  at  the  same  time  it  is  obvious 
that  his  experiments  were  not  adapted  for  arriving  at 
general  conclusions,  because  the  water  was  always  delivered 

I     upon  the  same  wheel ;  for  it  is  clear,  from  the  preceding 

■bivestigation,   that  every  particidar  wheel  must  have  its 

H^articular  maximum. 

"^  In  Smeafon's  experimenta  on  overshot  wheels,  the  wheel 
was  2  feet  in  diameter,  therefore  the  height  of  the  fall 
should  be  2^  feet.  Now  the  square  root  of  2^  is  1*5; 
and  1 -5  X  2-67  =  -t-005,  that  is,  the  velocity  of  the  wheel 
should  be  4,  feet  per  second ;  or  it  should  make  38  turns 
per  minute.  Smeaton  infers  that  "  the  best  velocitj'  for 
practice"  will  be  when  a  wheel  of  2  feet  tliameter  makes 
30  revolutions  per  minute.  (Miscellaneous  Papers,  p.  51.) 
But  his  model  had  much  more  friction  in  proportion  to  the 

I  effective  force  of  water  on  the  wheel  than  two  thirds,  that 
fe  have  here  c-alculated  upon.     When  the  calculation  is 


! 


390  ON  rHE  VELOCITT  [eIIATT. 

made  aooordiiig  to  the  fricti<m  of  Smeaton's  model,  vm 

9r4i  s/  h ;  and  the  velodly  of  the  model  wheel  would  oome 
out  3*6  feet  per  second,  or  34  turns  per  minute.  Thk 
yelocity  will  perhaps  apply  correctly  enough  to  overshot 
wheels,  where  the  water  flows  on  at  the  summit,  and  to 
rough  made  machinery ;  but  the  former  calculation  is  that 
which  I  consider  most  correct,  for  the  improyed  kind  of 
wheels  here  pointed  out  It  is  to  be  understood,  that  the 
friction  allowed  for,  includes  all  the  kinds  of  resistance  and 
loss  of  force  which  lessen  the  useful  effect,  as  well  as  the 
resistance  of  the  rubbing  sur&ces,  properly  called  fiictka. 
Many  readers  will  think  that  two  thirds  of  the  effective 
force,  IB  greatly  too  much  to  be  lost  \  it  will  be  well  if  it 
draw  their  attention  to  lessening  the  stress  on  every  part 
of  machinery,  and  to  the  importance  of  having  few  rubbing 
surfeces,  and  other  causes  of  resistance. 

414.  On  computing  the  power  of  overshot  water-whedi. 
— ^In  determining  the  proportion  of  the  radius  of  the  wheel 
to  the  height  of  the  fall,  an  equation  is  given  for  the  eflec- 

tive  force.    Resuming  that  equation,  we  have  i  b  ( -) 

=  the  effective  force  of  the  water,  and  i  J^;  ( —\  = 

its  mechanical  power.  But  the  quantity  of  water  expend- 
ed in  maintaining  this  power,  will  he  bv.  Hence,  the 
quantity  of  water  expended,  is,  to  its  mechanical  power,  as 

1  :  i  (1=^). 

When  the  wheel  is  supplied  at  the  summit,  j:  =  ^  c ; 
and  therefore,  the  quantity  of  water  expended,  is  to  its 
mechanical  power,  as  1  :  ^  c.  Or  the  power  is  equal  to 
half  the  weight  of  water  supplied  to  the  wheel. 

The  same  relation  takes  place  when  jr  =  o ;  that  is, 
when  the  wheel  is  supplied  at  the  height  of  the  axis.  Hence 
when  the  radius  of  a  breast  wheel  is  equal  to  the  effective 


APPEND.]  OF   WATER-WHEELS.  SSt\ 

height  of  the  Fait,  its  power  will  be  the  same  as  that  of  am, 
overshot  wheel  supplied  at.  the  summit. 

Whcu  the  wheel  is  supplied  at  the  point  which  pro- 
dooes  the  greatest  effect,  x  =>  -HQ-Jit  c ;   and  consequentlv 
the  quantity  of  water  expended  is  to  its  mechanical  power 
as  1  :  0*5857  c.     This  effect  is  greater  than  when 
wheel  is  supplied  at  the  sxunmit  in  the  ratio  of  1*1714 

These  comparisons  will  convey  some  useful  infonuatioD 
to  many  readers  ;  and  they  may  sometimes  suggest  to  scien- 
tific wtiters  the  advantage  of  studying  the  actual  nature 
of  machines  i  for  relations  so  extremely  obvious  and  simple 
could  never  have  been  overlooked  by  any  one  ivho  might 
have  condescended  to  examine  the  subject. 

The  power  of  a  water-wheel  may  be  considered  under 
two  points  of  view  j  each  of  which  has  its  peculiar  use. 
If  we  wish  to  compare  it  mth  any  other  first  mover,  then  we 
shall  have  to  calculate  its  mechanic^  power.  But  when 
it  is  desirable  to  compute  the  resistance  it  will  overcome  at 
the  working  point,  the  effective  force  should  bo  calculated. 

415.  "When  the  water  flows  upon  the  wheel,  either  at  or 

above  the  axis,  the  mechanical  power  is  ^  be — -  cubic 


wer       { 

ion  ^\ 


4 


bs.    \Vhen  bv  a  the 
quantity  of  water  expended  in  a  second,  in  cubic  fcpt 
the  part   of  the  circumference  between  the  lowest 
of  the  wheel,  and  the  place  where  the  water  flnwe 
it  in  feet,  and  j-  the  part  o[  the  eireiMfetw^,..  i^-_ 
the  point  which  is  level  with  the  axic,  and  that  »K-,„    i^ 
water  flows  upon  the  wheel  in  feet. 

Throughout  these  Esvvs,  the  M«»-lnMf^  ■«»»•  «/ 
horse  is  estimated  at  300  tU.  HKnia^  wiA  a  JtUtlu  <rf  sA 
feet  per  second.     Then  a  water-rind  wiQ  fce^^j 
31-33  ip(c'-gjO  00*86^  i-^.^^^"^ 


332  ON   THE   VELOCITY  [ESSAY  V. 

When  the  water  flows  on  either  at  the  summit  or  at  the 
level  of  the  axis,  the  mechanical  power  is  31*25  bvclbR.  or 
it  is  =  0*00426  bvc  horses. 

When  the  water  flows  on  at  52f  degrees  distant  from 
the  summit,  the  mechanical  power  is  37*192  bvclhs.  or  = 
•005  bvc  horses.  Since  in  this  case,  c  =  127i  degrees  of 
the  circumference,  we  have  c  =  127^  x  '0174533  r ;  and 

as  r  =  *554  h ;  and  v  =  2*67  >/  A ;  hy  substituting  these 
quantities,  we  have  122*176  bhi  lbs.  =  the  mechanical 
power ;  or  *0l64  bhi  =  the  number  of  horses,  where  h  = 
the  whole  height  of  the  fall  in  feet,  and  b  the  area  of  the 
apertiure  through  which  the  water  flows  upon  the  wheel  in 
feet. 

« 

416.  The  effective  force  is  31*25  be Iha.  when  the  water 
flows  on  either  at  the  summit,  or  at  the  level  of  the  axis. 

When  the  water  flows  on  at  52f  degrees  distant  from 
the  summit  of  the  wheel,  the  effective  force  is  37*192  iclbs. 
or  45*746  6  A  lbs. 


OF   THE   POWER  OF   BREAST   WHEELS. 


417.  When  the  water  flows  on  below  the  level  of  the 
axis  of  the  wheel,  it  may  be  termed  a  breast  wheel. 

Let  1/  be  the  distance  below  the  axis  measured  on  the 

circumference,  then  5-7 — 7 — r  equal  the  mechanical  power 

,.    /.        /.  31-25 c^fti;,,         „^ 

m  cubic  leet  01  water,  or lbs.     When  y  =  c  the 

c  -^-y  ^ 

power  will  be  reduced  one  half,  and  when  y  =  2  c  it  will  be 
reduced  two  thirds,  and  so  on. 

If  we  assume  that  the  mechanical  power  of  an  imdershot 
wheel  is  half  that  of  an  overshot  one  "  under  the  same  cir- 
cumstances of  quantity  and  fall ;"  *  then  it  will  be  an  ad- 

*  Smeaton's  Experiments,  Miscellaneous  Papers,  p.  49. 


APPEND.^  OF  WATER  WHEELS.  333 

yantage  to  employ  an  undershot  wheel  whenever  the  fall  is 
less  than  three  tenths  of  the  radius  of  the  wheel.  But 
since  the  radius  of  the  wheel  may  in  many  cases  he  dimi- 
nishedf  it  does  not  appear  to  he  desirahle  to  employ  an 
undershot  wheel  in  any  case,  except  where  the  quantity  of 
water  is  great  and  the  M  inconsideraiae. 


ESSAY   VI. 


ON 


CHANGING  THE  VELOCITY  OF  MACHINERY 


WHILE    IN    MOTION. 


INTRODUCTION. 

The  machinery  employed  in  manufactures  may  be  diyided 
into  two  classes :  1st.  Millwork.  2nd.  Smaller  Machinery. 
The  mechanism  described  in  this  Essay  belongs  to  the 
latter  class,  and  has  hitherto  been  chiefly  used  in  cotton- 
mills  ;  but  useful  hints  may  perhaps  be  taken  for  its  apph- 
cation  to  other  valuable  purposes. 

It  would  be  satisfactory  to  be  able  to  record  the  names 
of  the  inventors  of  many  of  the  ingenious  contrivances 
which  are  described  in  these  Essays.  But  the  secrecy 
which  interest  prompts  in  the  machinery  used  in  manufac- 
tures— the  same  difficulties  giving  rise  in  different  minds, 
without  any  communication  of  ideas,  to  the  same  means  of 
overcoming  them,  and  the  very  gradual  steps  by  which  im- 
provements are  usually  made,  render  it,  in  most  cases,  im- 
practicable to  trace  the  inventions  to  their  true  sources. 

It  may  be  taken  for  granted  that  the  reader  is  acquainted 
with  the  common  modes  of  altering  the  velocity  of  any  par- 


[8SAY  VI. j       CHANGING  THE  VELOCITY  Or  MACHINERY.    335 

liar  part  of  machinery,  by  changing  the  wheels  or  pul- 
This  change,  however,  requires  that  the  macliincry 
B  stopped  for  some  time  until  the  alteration  he  made. 

t  many  cases  occur  in  which  it  is  desirable  to  change 
velocity  without  such  loss  of  time.  Some  of  those 
s  shall  now  he  considered,  beginning  with  one  of  the 
;  simple, — that  of  changing  the  speed  of  a  turning- 
he,  according  as  the  nature  of  the  substance  to  be  turned, 
s  diameter  may  require. 


LATHS  MOTIONS. 


4.19.  A  series  of  pidleys  gradually  increasing  in  size 
«n  an  axle,  is  moved  by  the  mill,  and  on  the  spindle  of 
the  lathe  is  a  similar  series,  but  in  an  opposite  order,  so 
that  the  same  length  of  belt  will  work  on  all  the  opposite 

tlleys,  according  to  the  speed  required. 
These  series  resemble  two  tnmcated  cones,  having  the 
aller  diameter  of  the  one  opposite  the  greater  diameter 
of  the  other,  so  that  the  same  belt  is  equally  tight  on  what- 
ever pair  of  pulleys  it  may  work. 

This  contrivance  is  represented  by  Fig.  1,  Plate  XVI. 
applied  to  the  spindle  a  b  of  a  turning  lathe,  c  d  is  part 
of  a  shaft  driven  by  the  mill  at  a  certain  regular  velocity. 
When  a  slow  motion  is  required,  the  belt  works  at  ef; 
when  a  greater  velocity  is  required,  the  belt  is  shifted  by 
issing  it  to  one  side,  to  another  pair  of  opposite  puUeys. 
I  is  hoped  that  the  figure  will  make  this  so  clear,  that  all 
her  explanation  will  be  unnecessary. 

OBSERVATION. 
(420.  This  contrivance,  very  simple  in  its  construction, 
I  found  of  important  practical  use  in  the  turning  of  va^ 
i  substances. 


336  ON    CHAN6IK6   THE  [E88AT  VI. 


II.— ALTEBNATB  0ONB8. 


421.  There  is  another  contrivance  on  similar  principles 
to  that  above  described,  which  has  b^n  found  very  useful 
where  a  motion  constantly  varying  is  required.  Instead 
of  the  opposite  series  of  pulleys,  there  are  two  opposite 
cones. 

The  one  of  these  cones  gives  motion  to  the  other  by  a 
belt  which  by  the  machinery  is  gradually  moved  firom  one 
end  toward  the  other  of  the  cones. 

This  piece  of  machinery  is  represented  by  Fig.  2.  ab 
is  the  belt;  c  is  the  guide,  which,  receiving  its  motion 
from  the  machinery,  traverses  the  belt  at  pleasure,  mih 
any  velocity  which  the  case  may  require. 

Thus  the  one  cone  moving  at  a  uniform  motion,  com- 
municates a  varying  velocity  to  the  other. 


OBSERVATION. 


422.  This  piece  of  mechanism,  remarkable  for  its  sim- 
plicity, I  have  had  occasion  to  put  extensively  in  practice, 
and  have  found  it  give  great  satisfaction. 


III.<^-ALTERATION   OP   VELOCITY   BY    WHEELS  MOVING  ONE   ANOTHER 

BY   FRICTION. 

423.  The  same  eflTect  as  the  alternate  cones  is  some- 
times produced  by  the  rim  of  one  wheel  moving  on  the  face 
of  another,  by  means  of  the  roughness  of  their  surfaces, 
the  inequalities  of  which  may  be  considered  as  indefinitely 
small  teeth. 

Thus  A  B,  Fig.  3,  is  a  face- wheel,  moving  at  a  uniform 
rate. 

CD  is  another  wheel,  which,  from  the  face  of  ab,  re- 


lAY  VI.]  VELOCITY    OF    MACHINERV.  337 

ceives  a  vertical  motion.  Accordingly  as  it  is  required  to 
move  CD  slower  or  faster.  It  is  by  proper  contrivances 
ide  to  act  nearer  or  further  from  the  centre  of  a  b. 


f 

W4A 


OBSERVATIONS. 


42'i.  As  there  must  a  twisting  motion,  similar  to  that 
of  edge-stones  for  bruising  various  substances,  take  place 
here,  and  as  it  is  only  properly  applicable  to  cases  in 
which  the  strain  is  exceedingly  small,  1  apprehend  that 
the  alternate  cones  is  a  much  more  perfect  manner  of  pro- 
ducing a  change  of  velocity.  These  wheels,  however,  work 
verj'  well  for  regulating  the  taking  up  motions  of  the  bob- 
bins in  machines  for  ro\'ing  cotton  by  spindles.  In  this 
;,  the  force  required  is  verj'  small. 


Certain  eases  in  practice  require  an  instantaneous 

of   velocity ;    as,  for  instance,    when    those  ma- 

ines  for  spinning  cotton,   called  Alules,  are  moved  by 

Irer. 

jThc   mule  is  a  machine    different   in  its  construction 

\  that  brought  to  a  high  state  of  perfection  by  Sir  R. 

kwright. 

The  mule  is  better  adapted  than  the  water-ficist  frame 
(Arkwright's  Machine)  for  spinning  all  kinds  of  weft,  and 
produces  finer  yam  than  can  be  spun  by  any  other  machine. 
For  the  invention  of  the  mvk  we  are  indebted  to  James 
Crompton,  formerly  of  Hall-in-the-Wood,  near  13oltun-Ie- 

IDors,  Lancashire,  This  machine  was,  for  many  years, 
rked  by  hand  only,  the  variety  of  its  movements  render- 
f  it  difficult  to  accomplish  the  moving  of  it  by  power  of 
ter  or  of  steam  sufficiently  simple  to  be  of  common  use. 


338  ON    CHANGING   THE  [[eSS^T  VL 

William  Kelly  at  Lanark,  early  obtained  a  patent  for  a 
mode  of  working  this  machine  by  power,  but  it  was  not 
until  a  considerable  time  afterward  that  power  was  gene- 
rally adopted.  The  plans  which  were  tried  were  very 
various,  and  the  improvement  was  progressive.  One 
happy  consequence  of  this  improvement  has  been  expe- 
rienced ;  the  spinners  are  now  found  to  enjoy  better  health 
than  they  did  when  they  had  to  labour  hard^  while  they 
breathed  in  warm  and  confined  apartments. 

In  order  to  save  time  after  the  carriage  of  the  mule  is 
brought  to  its  furthest  extent,  it  is  necessary  to  increase 
the  velocity  of  the  spindles.  This  increase  of  velocity  is 
called  the  double  speed.  Various  contrivances  have  been 
adopted  for  this  purpose,  but  three  only  shall  be  described; 
one  being  performed  by  r(^eSf  another  by  belts^  the  third 
employing  the  aid  of  wheels.  This  last,  indeed,  firom  its 
greater  certainty,  is  jvhat  is  most  generally  adopted.  These 
contrivances  will,  however,  serve  to  shew  the  progress  of 
improvement  in  this  species  of  machinery. 


DOUBLB   SPBBD. 


First  Canstniction. 


426.  The  axle  ab,  Fig.  4,  Plate  XVII.  is  suspended  by 
a  cast  iron  frame  from  the  ceiling  of  the  room.  This  axle 
is  kept  in  motion  by  means  of  the  fixed  pulley  c,  which  is 
moved  by  a  belt  from  the  mill- work.  On  the  same  axle  are 
two  loose  pulleys  d  and  e.  (Essay  IV.  Art.  280.)  Ropes 
from  these  pulleys  communicate  with  the  fast  pulleys  f  and 
G,  on  the  axle  x  y  of  the  fly  wheel  of  the  mule. 

The  loose  pulleys  have  catches  on  their  sides;  while 
these  are  disengaged  the  mule  is  at  rest.  In  order  to  put 
the  mule  in  motion,  the  smaller  pulley  e,  by  means  of  the 
sliding  guide  ikl,  is  slipped  to  one  side,  so  as  to  lay  hold 


I 


;AV  VI.]  VELOCITY    OF    MACHINERY.  339 

of  the  glfuid  H,  which  is  fixed  on  the  axle,  and  carries  the 
pulley  round  along  with  it,  and  thus  moves  the  mule  at  its 
slower  motion. 

When  the  fly  wheel  w  has  made  a  suflScient  number  of 
revolutions  at  this  rate,  the  slider  is  moved  by  peans  of 
wheel-work  and  a  wiper  toward  the  fast  pulley  a,  which 
motion  disengages  the  small  pulley  from  the  gland,  and 
engages  the  larger  pulley  with  c,  which  produces  a  quick 
motion  in  the  fly  wheel. 


OBSERVATIONS. 

4^.  This  apparatus  was  in  use  in  Manchester  in  the 
jear  1797>  but  the  shocks  produced  by  the  catches  (Essay 
Art.  281.)  and  other  imperfections,  soon  occasioned 
disuse.  But  there  is  often  much  to  be  learnt  from  the 
examination  of  machines  which  have  been  abandoned.  It 
is  but  by  comparison  of  things  of  the  same  species  that  we 
are  able  to  appreciate  their  true  value. 


b 


DODBLB   BPEBD. 


Second  ConOruction. 


428.  This  apparatus  differs  from  the  First  Construction, 
principally  in  having  belts  instead  of  ropes  for  communi- 
cating motion. 

On  the  axle  a  b,  there  are  five  pulleys,  e,  d,  c,  g,  h,  all 
of  them  loose  but  c,  which  is  fast.  When  the  belt  from 
the  mill-work  is  on  c,  the  mule  is  at  rest,  because  the 
axle  revolves   without  carrying  round  any  of  the  loose 


In  order  to  put  the  mule  in  motion,  the  belt  is,  by  means 
of  a  sliding  guide,  shifted  on  to  the  pulley  d,  which  carries 


840  ON   CHANGING   THE  [^£88AT  VI. 

round  the  pulley  x  along  with  it ;  and  by  another  belt» 
moves  the  pulley  f  on  the  fly  wheel  axle  xy,  and  thus 
moves  the  mule  at  its  slower  motion ;  afterward  (as  was 
described  of  the  First  Construction)  the  sliding  guide 
shifts  the  belt  from  d  to  o,  which,  by  carrying  round  h  in 
a  similar  manner,  produces  a  quick  motion  in  the  fly 
wheel  w. 

OBSERVATION. 

429*  This  construction  was  in  use  in  Manchester  m  the 
year  1799 ;  and  as  the  shocks  complained  of  in  the  First 
Construction  did  not  occur  in  this,  it  was  found  a  material 
step  in  the  improvement  of  working  mules  by  power. 

DOUBLE   8PBBD. 

Third  CanstructUm. 

430.  This  construction  differs  from  the  second^  in  having 
the  whole  apparatus  attached  to  the  framing  of  the  mule, 
and  in  having  the  aid  of  toothed  wheels  for  producing  the 
change  of  velocity. 

On  the  axle  a  b,  Fig.  6,  are  three  pulleys,  c,  d,  e.  The 
pulley  c  is  fast  on  the  axle,  d  and  £  are  loose,  but  on  the 
side  of  E  is  fixed  the  small  spur-wheel  f.  The  larger  spur- 
wheel  G  is  fast  on  the  axle  ab. 

On  the  axle  x  y  are  fixed  other  two  spur-wheels,  h  and 
I,  of  the  same  size  as  those  on  the  axle  a  b,  but  placed  so 
as  that  the  larger  wheel  on  the  one  axle  shall  be  constantly 
in  gear  with  the  smaller  on  the  other. 

When  the  belt  (put  in  motion  by  the  mill-work)  is  on  d, 
the  mule  is  at  rest ;  when  shifted  on  to  e  it  carries  the 
smaller  wheel  f  round  with  it,  which  being  in  gear  with 
the  larger  wheel  h,  moves  the  fly  wheel  axle  x  t  at  its 
slower  motion. 


S88AT  VI.3  VELOCITY   OF   MACHINERY.  341 

On  the  other  hand,  when  the  belt  is  shifted  to  the  pul- 
ley Cy  which  is  &st  on  the  axle,  it  carries  round  the  larger 
wheel  G,  which  is  also  fast,  o  being  in  gear  with  the 
smaller  wheel  i,  moves  the  fly  wheel  at  the  greater  velocity, 
or,  as  it  is  termed,  at  double  speed. 


OBSERVATIONS. 

431.  This  piece  of  mechanism  was  first  adopted  in  Man- 
chester about  the  year  1800,  and  although  sometimes  its 
parts  may  be  somewhat  differently  arranged,  it  continues, 
I  believe,  still  in  general  use. 

While  it  is  firee  from  the  shocks  produced  by  catches,  it 
is  also  (owing  to  having  the  change  of  velocity  produced 
by  wheels)  free  from  the  uncertainties  of  motion  arising 
from  any  change  in  the  tightness  of  the  belts  as  employed 
in  the  Second  Construction. 


ESSAY    VII. 


ON  THE 


FRAMING  OF  MILL-WORK. 


PREFACE. 

The  preceding  Essays  relate  principally  to  the  moving 
parts  of  macliinery,  but  as  it  seemed  essential  to  a  system 
of  mill- work,  to  say  somewhat  on  the  subject  of  the  Jram- 
ing  which  supports  the  moving  parts,  Buchanan  was  in- 
duced to  commit  to  paper  the  following  ideas  on  that  head. 


SECTION  I. 

432.  The  general  principles  of  carpentry  must  obviously 
be  applicable  to  the  framing  of  mill-work.  These  prin- 
ciples I  shall  not  here  repeat,  but  beg  leave  to  refer  to 
what  Professor  Robison  has  written  on  this  subject,  in  the 
Encyclopaedia  Britannica,  and  to  Mr.  Peter  Nicholson's 
various  writings  on  the  subject;  I  shall  here  consider  only 
the  peculiarities  of  the  framing  of  mill- work*. 

433.  Mill-work,  from  its  motion,  occasions  a  tremor  on 

*  See  also  Art  Carpentry,  New  Supplement  to  the  Encyclopsedia  Bri- 
tannica. Tredgold's  Elementary  Principles  of  Carpentry,  4to.  1820;  and 
Practical  Essay  on  Cast  Iron,  Svo.  1822. 


lESSAY  VII.3         ON    THE    FRAMING    OP    MILL-WORK.  343 

ail  the  parts  of  its  framing,  which  subjects  it  to  much  more 
speedy  decay  than  the  mere  pressure  upon  carpentry. 

Besides  this  general  tremor,  it  is  often  subjected  to  vio- 
lent sudden  thrusts,  from  the  bad  action  of  the  wheels,  or 
from  reciprocating  motions. 

It  ought,  therefore,  not  only  to  be  sufficiently  strong  and 
stiff,  but  sufficiently  heavy,  to  give  solidity  and  steadiness. 

Where  the  framing  of  machinery  is  not  firm  and  well 
bound,  a  ^ibratorj'  motion  in  its  parts,  of  course,  takes 
place ;  which  vibratory  motion  expends  a  considerable  por- 
tion of  the  power  applied.  This  loss  of  power  ia  very  diffi- 
cult of  investigation.  It  is  certain,  however,  that  whatever 
motion  of  a  vibratory  nature  is  communicated  to  the  fram- 
ing and  objects  in  contact  with  it,  (absfracting  from  the 
elasticity  of  the  parts,)  must  be  lost  to  the  effect  the  ma- 
chine would  produce,  were  the  parts  sufficiently  strong  and 
well  bound  together ;  and  it  is  to  be  observed,  that  firm 
and  well-bound  framing  is  much  preferable  to  heavy  fram- 
ing not  so  well  connected  in  its  parts.  It  is  as  certain, 
that  though  the  framing  in  either  case  may  be  constructed 
so  as  to  be  equally  strong ;  yet  the  heavy  framing,  from  its 
vibration,  will  expend  more  of  the  original  power  than  that 
which  is  less  heavy  but  firmly  connected. 

434-.  Besides  strength,  siiffnessy  and  solidity,  the  framing 
of  mill-work  requires  to  be  constructed  so  as  to  be  e«.«y  of 
repair ;  and  so  contrived,  that  any  particular  part  may  be 
reftaired  or  renewed  with  the  least  possible  derangement  to 
the  other  parts  of  the  framing. 

435.  There  is  another  circumstance  in  this  species  of 

ing  which  demands  great  attention.     The  shajis  often 

tquire  to  be  restored  to  Ihetr  true  situations,  from  which 

*  ihev  may  have  deviated  by  the  wearing  of  the  parts.    Now 

the  framing  ought  to  be  so  constructed  as  easily  to  admit 

of  this  restoration  of'  (he  shttjis,  as  also  of  any  other  shift- 

Liiig  of  them  which  may  in  practice  become  ncccseary. 


344  OK  THE   FRAMING   OF   MILL-WORK.      [E88AT  VH. 

436.  But  thougli  the  framing  which  supports  the  parts 
of  mills  and  machines  should  be  firm,  it  is  an  advantage 
that  the  part  on  which  any  axis  rests  should  have  a  small 
degree  of  elastic  tremor  when  the  machine  is  in  motion. 
Such  tremor  has  considerable  power  in  diminiRhing  the 
friction.  It  may  further  be  observed,  that  framing  to  sup- 
port  machinery  should  be  as  independent  of  the  building 
as  possible,  because  the  tremor  it  always  communicates  is 
exceedingly  injurious. 

Before  proceeding  further  into  the  subject,  it  may  be 
proper  to  consider  the  bearings  of  shafts. 

SECTION  11. 

OP  THB  BEARINGS  OF  SHAFTS. 

437-  The  bearings  on  which  gudgeons  and  journals  rest 
and  revolve,  are  sometimes  termed  Pillows^  and  fi^uently 
BrasseSf  from  being  often  made  of  that  substance. 

The  bearings  for  pivots,  at  the  lower  extremity  of  up- 
right shafts,  are  denominated  Steps ;  the  parts  where  the 
journals  of  vertical  shafts  or  spindles  turn  and  bear  against 
are  called  Bushes;  and  for  small  spindles,  such  as  those 
used  in  the  manufactures  of  flax  and  cotton,  Breasts. 

It  has  become  general  to  fix  pillows  in  blocks  of  cast 
iron.  Hence  the  term  Pillow  Blacky  and  sometimes,  cor- 
ruptly. Plumber  Block.  In  Manchester  they  are  called 
Pedest€ds. 

438.  The  substances  used  for  Pillows^  &c.,  are  various, 
but  brass  is  the  most  common*.  Other  substances,  how- 
ever, which  are  cheaper,  have  in  many  instances  been  found 
equal,  at  least,  in  durability. 

*  The  metal  our  author  terms  brass,  is  usually  the  composition  of  copper 
and  tin,  called  gun  metal.  Gun  metal  is  much  harder  than  common  brass, 
and  much  more  durable.  Common  brass  is  a  compound  of  copper  and 
sine,  and  is  now  rarely  used  for  bearings. 


iSAy  VU.]       ON    THE    FRAMING    OF   MILL-WORK.  345 

At  the  cotton  works  of  Deanston,  near  Down,  a  water 
wheel  has  nm  nearly  30  years  on  pillows  of  cast  iron,  with 
little  sensible  wear  on  the  gudgeons,  nor  were  they  ever 
found  liable  to  heat". 

The  outer  skin  of  cast  iron,  particularly  when  caat  in 
metallic  moulds,  is  remarkably  hard,  and  it  is  reasonable 
to  suppose  that  it  would  make  a  durable  pillow,  as  we  have 
|seen  is  the  case  in  the  above  instance. 

Mr.  Murray  of  Leeds  was  enabled  to  bore  the  hardest 
1st  iron,  some  cylinders  of  which,  from  the  whiteness  of 
he  grain  at  the  places  broken  off  by  a  chisel,  denoted  its 
fuperior  quality.     Such  iron  is  equally  hard  throughout. 

A  patent  was  granted  long  ago  for  wheel  bushes  of  me- 

,  of  a  peculiar  hardness,  which  proved  to  be  nothing 
'  more  than  very  hard  cast  iron,  but  the  patentee  had  dis- 
covered a  mode  of  boring  it,  which  Murray  imitated. 

Stone  has  often  been  used  with  good  effect  for  pillows  for 
gudgeons  and  journals.  The  great  objection  to  stones  for 
this  purpose,  is  the  difficulty,  arising  from  their  hardness, 
of  forming  them  into  proper  shapes. 

At  Sheffield,  where  the  joumejTnen  grinders  are  obliged 
to  keep  this  part  of  their  machinery  in  oil,  and  in  repair, 
they  liave  found  from  long  experience,  that  a  piece  of  green 
(unseasoned)  thorn  tree  is  exceedingly  durable  f .  But  in 
general  they  prefer  using  brown  paper,  adding  always  one 
ply  some  time  after  another,  so  as  to  form  a  kind  of  paste- 
board ;  this  substance  they  find  less  liable  to  heat,  and 
much  more  durable  than  brass  t. 

•  Heating  geiieraUy  lakes  place  from  the  surfaces  of  the  journal  and 
pillow  being  too  small,  and  sometinieB  from  the  journal  having  worn  too 
deep  into  the  pillow,  in  which  last  ease,  in  jiarticular,  a  great  friction  takes 
jtloce. 

t  Qreen  oak  soaked  in  boiling  oil,  is  said  to  be  need  with  advantage, 
r's  Additions  to  Ferguson,  vol.  ii.  p.  179. 

}  If  the  bearings  for  gudgeons  were  made  hy  screwing  maay  thickneases 
of  posteboftril  together,  in  the  same  manner  as  the  rollers  of  calendars  are 
iDule,  they  would  be  extremely  durable,  and  have  very  little  fricdon. 


346  ON   THE   FRAMING    OF   MILL-WORK.       [^£S8AT  VII. 

Wooden  pillows  ore  often  used.  Box  wood  and  lignum 
yitsB  were  long  in  use.  The  latter  ha^  been  found  an  im- 
proper substance  for  the  purpose.  Beech  is  preferable  to 
either,  and  has  been  used  with  great  success  for  steps. 
Holly  has  also  been  found  to  answer  well  for  the  same 
purpose*. 


• 

FOBMS  OP  8TBP8  AND  OP  PIVOTS  OP  UPBIOHT   8HAPT8. 


4<S9«  The  most  usual  form  is  to  terminate  vertical  shafts 
somewhat  like  an  egg^  leaving  the  pivot  rather  smaller  up- 
wards, to  prevent  it  from  binding  in  the  step.  (Plate 
XVIII.  Fig.  1.) 

440.  Another  mode,  (see  Fig.  2,)  is  to  terminate  the 
shaft  flat  at  the  lower  end,  and  make  a  small  oblong  recess 
in  it.  To  this  end  is  fitted  a  round  piece  of  steel  a,  flat 
below,  excepting  a  groove  a  across  it,  something  like  that 
in  the  head  of  a  screw  naiL  The  use  of  the  groove  is  re- 
gularly to  feed  in  the  oil.  The  step  into  which  the  pivot 
works  is  made  square  or  octagon,  so  as  to  supply  oil  from 
the  angles.  Nos-  2,  3,  and  4,  Fig.  2,  are  diflFerent  views  of 
the  piece  of  steel  a. 

441.  A  pivot  and  step  on  this  construction  was  deposited 
with  the  Society  for  the  Improvement  of  Arts,  and  was 
said  to  have  been  20  years  in  use  in  a  horse-mill,  without 
being  perceptibly  worn.  Buchanan  applied  this  contriv- 
ance in  a  case  of  heavy  upright  shafts,  where  other  means 

*  Frictioii  rollers  are  sometiiiies  employed  to  diminish  the  quandtj  of 
friction,  bat  not  with  much  adyantage  in  heavy  machinery,  becaase  they  are 
liable  to  get  out  of  order,  and  require  very  accurate  workmanship.  Their 
effect  depends  on  converting  a  sliding  into  a  rolling  motion,  with  only  the 
slide  of  the  small  axis  which  retains  the  roller  in  its  place.  Friction  wheels 
arc  also  used  for  the  same  purpose ;  they  differ  from  rollers  in  this — the 
stress  on  the  axis  is  borne  by  the  axes  of  the  friction  wheels,  but  the  stress 
is  on  the  surface  of  the  rollers.  The  advantage  of  finctioii  wheels  is  very 
trifling. 


AY  VII.J      ON    THE   FRAMING    OF    MILL-WORK. 


347 


3  found  inadequate  to  prevent  the  action  of  the  pivot, 
t  it  ever  after  gave  satisfaction  in  its  use.  It  is  to  be 
wn'ed,  however,  that  it  does  not  answer  well  in  cases 
I  much  lateral  pressure,  and  a  better  mode  (see  Fig. 
J  is  now  in  use  in  several  mills.  The  steel  pivot  is  made 
Cylindrical,  and  fixed  into  the  foot  of  the  upright  shaft. 
The  bottom  of  the  step  is  convex,  and  there  is  interposed 
one  or  more  pieces  of  steel  formed  Uke  a  double  convex 
lens.  These  pieces  of  steel  having  a  little  motion,  make 
the  relative  motion  of  the  pivot  less,  and  consequently 
ien  the  friction. 

442.  The  egg-formed  pivot  is  sometimes  made  separate, 
wrought  iron  and  steel,  and  inserted  into  the  lower  end 
the  cast  iron  shaft*. 

443.  The  late  Mr.  Bramah,  in  the  specification  of  his 
tent  machinery  for  surface  planing,  includes  a  mode  of 
Oning  pivots  entirely  on  a  fluid,  and  raising  or  depressing 

□  at  pleasure,  by  means  of  a  small  forcing-pump  and 
>  cock.     See  Gregory's  Mechanics,  vol.  Ji.  p.  418,  2d 
Ltion. 

444.  The  journals  of  upright  shafts  are  supported  some- 
les  by  breasts,  (Fig.  4,)  and  sometimes  by  bushes,  in 
ising  through  a  floor. 

445.  The  spindles  of  millstones  usually  run  in  wooden 
hes.  A  block  of  cross  elm,  abc,  Fig.  5,  about  9  inches 
meter,  and  3  inches  thick,  forms  the  principal  part  of  it, 

d  is  lodged  in  the  eye  of  the  millstone.  In  order  that 
}  spindle  may  at  all  times  run  steadily,  there  are  three 
Kes  of  hard  wood,  d,  d,  d,  lot  into  grooves  in  the  block, 

'  The  rubting  Burfaces  of  caat  iron  pWota  ahouJd  not  have  a  greater 
Bore  upon  them  than  one  ton  QpoQ  a  square  inch,  or  they  vill  he  very 
ject  to  heat,  and  the  friction  and  weai  will  he  increased.  Large  vertical 
^  may  often  be  made  to  revolve  on  conical  rollers,  on  the  same  prin- 
e  aa  fnctiDn  rollers;  when  they  nro  well  made,  the  motion  is  very 
and  the  friction  much  reduced. 


346        om  TBE  PBAJOVG  or  xux^-wobk.       [bmat  vil 


•o  that  tbeir  three  ends  embraee  die  ^indies.  TheeepieoeB 
are  of  equal  breadth  throaghont,  to  that  thej  may  easify 
be  wedged  fiirwafd  when  dinr  wear.  This  flimple  and  in- 
genioiis  cootriraiioe  has  been  vor  long  in  use.  Some  use 
apiece  ci  cast  iron  in  {ireAsrence  to  die  Uock  ci  efan,  to 
answer  the  same  pivpose^  and  some  a  greased  ntpe  to  nm 
the  spindle  in,  instead  <rf  die  dnree  pieces  ci  wood. 


SECTION  m. 

09  WOODSH  FSAMniO. 


446.  The  framing  used  for  suppcnrting  die  gudgeons  of 
a  water  wheel  is  denominated  the  headstock  firaming. 

Af  Fig.  6,  represents  the  headstock,  which  omtains  the 
pillow-block,  BODE  is  the  frame. 

The  frame  is  supported  by  the  top  of  the  building  of  the 
arch  or  wheel-pit 

The  headstock  rests  on  the  frame,  and  is  moveable  hori- 
zontally on  all  sides ;  when  in  its  proper  situation,  it  is 
kept  there  by  a  key  or  wedge  g  at  each  end  of  the  head- 
stock,  which  is  supported  by  the  dovetail  form  of  the  lower 
])art  of  the  headstock. 


FRAMING   FOR  LYING  SHAFTS. 


447.  Lying  shafts  are  usually  supported  by  bridges  be- 
tweon  posts,  a  b.  Fig.  7,  represents  part  of  a  lying  shaft, 
c  1)  the  bridge  which  carries  the  shaft.  It  is  raised  or 
lowenni  by  means  of  a  key  below,  and  another  above  it,  in 
the  mortice  of  the  posts  eg  and  hi,  through  which  it 
passes. 

The  bridge  is  also  moveable  on  end,  and  is  kept  in  its 
pn>iH>r  situation  by  means  of  small  keys  k,  k,  k,  k. 


lAT  VII.] 


ON    THE   FHAMING    OF    MILL-WORK. 


3-1.9 


448.  The  posU,  instead  of  being;  each  made  of  one  solid 
piece  of  timber,  are  sometimes  framed  of  separate  pieces, 
as  represented  in  Fig.  8. 

LjTUg  shafts,  instead  of  being  supported  by  posts,  are 
sometimes  suspended  from  a  ceiling,  as  shewn  in  Fig,  9- 
The  bridge  is  tempered  by  keys,  &c.,  as  when  posts  are 
used. 


J   FOB   UPtllOHI   SHAFTS. 


449-  Upright  shafts  are  generally  supported  by  bridges 
adjusted  endwise,  and  upward  and  downward,  like  those  of 
the  lying  shaft ;  hut  in  order  that  they  may  be  moved  hori- 
zontally in  every  direction,  the  pedestal  is  contrived  to 
deceive  keys  at  its  ends,  similar  to  a  hcadstock.     (See 

Wt  '"'^ 

^Bocrews  arc  frequently  used  instead  of  wedges  for  adjust- 
ing the  step. 

450.  Fig.  1 1  represents  the  framing  of  an  upright  and 
^UDg  shaft  connected  by  bevelled  wheels. 
^■^1.  Sometimes  a  bridge  is  not  immediately  supported 
^Pposts,  but  by  intermediate  pieces,  which  are  called 
cJoves.  This  construction  is  common  in  single  corn-mills. 
Thus,  AB,  Fig.  12,  is  a  bridge;  cd  and  ef  arc  cloves. 

452.  Respecting  the  decay  of  timber,  and  the  means  of 
preventing  it,  Buchanan  refers  to  Dr.  Parry's  paper,  in 
the  Transactions  of  the  Bath  Agricultural  Society,  and  re- 
printed in  Nicholson's  Journal,  Vol.  XX.  Nos.  85,  86,  87» 
;uid  Repertory,  No.  LXIII.  That  paper  appears  well 
worthy  the  attention  of  those  who  have  occasion  to  con- 
struct works  of  timber*.  Kyan's  mode  of  preserving 
nber,  till  a  better  shall  be  proposed,  now  supersedes  all 
VschemoB  heretofore  promulgated. 

"  Blementdty  Principles  of  Carpeutiy,"  Sect.  X.  Art.  327— 


J 


350  ON   THE   FRAMING   OF   MILL-WORK.       [eMAT  YIL 


SECTION  IV. 


OF  CAST  IBON  FBAMINO. 


453.  In  a  preyious  part  of  this  work,  mention  lias  been 
made  of  the  great  increase  of  late  years  of  the  use  d 
cast  iron  in  mill-work. 

Cast  iron  possesses  great  superiority  over  timber,  for 
constructing  the  framing  of  mill-work.  It  is  not  only 
much  more  durable,  but  &t)m  the  uniformity  of  its  texture, 
may  be  converted  into  any  shape,  so  as  to  give  it  great  adU 
vantage  in  arranging  the  materials  with  respect  to  strength, 
and  proportioning  it  to  the  stress  it  has  to  sustain.  Tim- 
ber, on  the  other  hand,  being  stronger  in  some  directions 
than  others,  and  of  very  limited  breadth,  is  confined  in  its 
arrangement,  and  requires,  in  certain  cases,  much  work- 
manship ;  whereas,  after  the  patterns  for  cast  iron  are  once 
made,  any  number  of  castings  may  be  formed  from  them 
with  very  little  labour  or  trouble. 

Those  who  have  scientifically  considered  the  strength 
and  stress  of  materials,  know  that  when  timber  is  broken 
by  any  lateral  pressure*,  it  is  owing  in  a  considerable  de- 
gree to  the  compression  of  the  beam  on  the  hollow  side, 
which  puts  the  fulcrum  of  the  ideal  lever  much  nearer  the 
point  of  resistance  than  it  would  be  in  a  substance  less 
liable  to  compression.  Cast  iron  is  much  less  liable  to 
compression  than  timber,  which  gives  it  an  advantage  in 
withstanding  lateral  pressure,  greater  than  might  be  ex- 

*  See  Emerson's  Mechanics,  Sect.  VIII.  p.  93,  and  Gregory's  Mecha- 
nics, Vol.  I.  Book  I.  Chap.  V. 

These  references  must  have  heen  made  without  consulting  the  works 
quoted,  as  the  investigation  of  the  strength  of  heams  is  conducted  by  both 
these  writers  on  the  supposition  that  the  virtual  fulcrum  is  an  incompres- 
sible arris  at  one  of  the  surfaces  of  a  beam ;  and  the  one  conadere  the 
materials  to  be  extensible,  the  other  inextensible. 


lAT  VII.]       ON    THE    FRAMING    OF    MILL-WORK.  351 

■jed  from  the  mere  comparative  absolute  cohesion  of  the 
wtances'. 

54.  "  Iron    is   generally  much   more   uniform   in   its 

.  than  wood;  yet  experiments  shew  that  there  is 

i  difference  occasioned  by  different  kinds  of  ore  :  the 

ference  is  not  only  found  in  iron  from  different  furnaces, 

t  from  the  same  furnace  and  the  same  melting ;  this  may 

!  in  a  great  measure  from  the  different  degrees  of  heat 

,ch  it  has  when  it  is  poured  into  the  inould."t 

Banks  concludes  that  a  bar  of  the  weakest  cast 
,  1  inch  square,  and  1  foot  long,  will  break  with  about 
.,  and  that  cast  iron  is  at  an  average  4  times  as 
strong  as  oak,  and  5^  times  as  strong  as  deal;  the  weight 
being  in  all  cases  applied  in  the  middle ;  the  beam  lying 
horizontally,  and  supported  by  props  J, 

■ls5G.  The  strength  of  any  beam,  to  withstand  any  weight, 
being  as  the  breadth  and  the  square  of  the  depth,  (see 
Essay  II.  on  the  Shafts  of  Mills,  &c..  Chap.  IV.  Emerson, 
p.  93.)  it  is  evident  that  a  bar  of  cast  iron  of  the  same 
length,  must  be  much  stronger  when  its  tranverse  section 
is  like  Fig.  14,  than  when  like  Fig.  13.  The  form  repre- 
sented in  Fig.  14  has  a  further  advantage,  that  of  greater 
stiffness.  The  distinction  between  strength  and  stiffness 
is  not  in  practice  generally  understood  or  attended  to.  This 
distinction  is  most  easily  comprehended  by  considering  their 
limits.  The  limit  of  strength,  is  well  known  to  be  frac- 
ture, or  breaking.  The  limit  of  stiflfhess,  is  Jiexure,  or 
bending.     Now  stifihess  increases  in  a  much  higher  ratio 

•  The  idea  that  the  yirtmil  fulcrum  is  nearer  the  compressed  side  in  cast 
iron  ihaa  in  wood,  when  the  pieces  are  similarly  strained,  is  at  best  an  na- 
aenion  without  a  proof,  either  from  theory  or  experience.  And  at  the  time 
oar  author  wrote,  the  resistance  of  cast  iron  to  compression  was  greatly 
uvemtad.     See  Esaay  on  Cast  Iron,  Art,  63. 

^^  t  On^ry's  Mechanics,  Vol.  I.  Art.  A.  190. 

^^L%  Banks  on  Power  of  Machines,  &c.,  p.  94. 


d5S  ON   THE   FRAMING   OF  MILL-WORK.      [eSSAT  TU. 

ratio  than  strength,  viz,,  as  the  cube  of  the  depth  ••  For 
example,  if  we  double  the  depth  of  a  beam,  we  increase  its 
stress  only  4  times,  whereas  we  increase  its  stiffness  8 
times.     (See  Essay  II,  on  the  Shafts  of  Mills.) 

457-  For  these  reasons,  the  advantage  is  evident  of 
making  cast  iron  framing  in  thin  broad  plates,  at  right  an- 
gles to  one  another,  instead  of  imitating  the  solid  forms  of 
wooden  framing.  This  practice  is  called  by  millwrights 
feathering.  The  plate  is  sometimes  on  one  side,  as  repre 
sented  in  Fig.  15,  and  sometimes  its  section  is  like  the 
letter  t,  see  Fig.  14,  the  whole  being  one  solid  mass. 

The  common  practice  in  making  cast  iron  framing  now, 
is  to  imitate  wooden  framing,  which  has  been  found  from 
experience  sufficiently  strong  in  giving  the  same  breadth 
and  depth  of  the  several  pieces  at  their  point  of  greatest 
stress.  Thus,  suppose  Fig.  16  to  be  the  section  of  the 
timber  at  the  place  of  greatest  stress,  the  section  of  the 
cast  iron  is  made  like  Fig.  17»  or  like  Fig.  18 ;  advantage 
is  also  taken  of  the  nature  of  the  material,  to  give  it  a 
breadth  varying  in  proportion  to  the  stress.  This  variation 
in  shape  is  not  always  in  practice  judiciously  done ;  by  at- 
tending to  what  is  said  in  Essay  II.,  Chap.  IV.,  the  mill- 
wright, it  is  hoped,  will  be  better  enabled  to  proportion  the 
parts  to  the  stress  they  have  to  sustain  t.  In  addition  to 
what  is  said  respecting  the  making  of  patterns,  in  Essay 
I.  on  the  Teeth  of  Wheels,  Buchanan  says  it  is  a  good 
practice  to  give  the  patterns  a  thin  coat  of  oil  paint ;  as, 
while  it  preserves  the  pattern,  the  paint  makes  it  rise 
more  freely  out  of  the  sand. 

*  Young's  Nat.  Phil.  Vol.  ii.  Art.  333. 

t  The  most  advantageous  forms  for  different  purposes  have  heen  con- 
sidered in  the  Practical  Essay  on  Cast  Iron^  Sect.  III.  and  IV.,  where  ex- 
tensive tables  of  the  strength  and  stiffness  of  cast  iron  will  be  found,  which 
may  frequently  save  the  millwright  much  trouble  in  calculation ;  for  he 
cannot  always  have  examples  of  the  same  construction  to  refer  to,  either 
executed  in  wood  or  iron. 


ESSAY  Vll.]      ON    THE    FRAMING    OF    MILL-WORK.  353 

458.  To  give  an  instance  of  this  variation  of  form  in  the 
framing  of  mill-work,  bridges  of  wood  for  sustaining  the 
shafts  are  usually  made  as  represented  in  Fig.  19,  those  of 
cast  iron  as  shewn  in  Fig.  20. 

4s59.  In  cast  iron  framing,  advantage  is  also  taken  of 
the  properties  of  the  hollow  cylinder,  of  the  economical 
ilication  of  which  form  in  nature  we  have  so  many  beau- 
tiful examples.     (See  Essay  II.) 

460.  A  headstock  of  cast  iron  for  a  water-wheel  is  re- 
presented in  Fig.  21,  Plate  XIX. 

461.  Various  modes  are  used  of  suspentling  shafts  from  a 
ceiling.  Fig.  22  represents  a  construction  in  very  general 
practice. 

462.  Fig.  23  represents  the  cast-iron  framing  of  a  flour 
mill  having  three  pair  of  mUl-stones,  and  to  Fig.  24',  a  ma- 
chine used  in  bleaching,  called  Squeezers. 

After  the  process  of  washing  by  the  dash-whoel,    the 
iter  is  compressed  from  the  cloth  by  means  of  this  ma- 


Kfiluni 
■       So 


Squeezers  consist  of  a  pair  of  wooden  rollers,  which,  in 
mo\Tng,  draw  the  cloth  through  between  them.  The  lower 
roller  receives  its  motion  from  a  mill,  and  the  uppermost 
is  pressed  down  upon  it  by  means  of  levers.  Till  of  late 
these  rollers  were  fixed  in  strong  wooden  frames ;  hut  the 
framing  is  now  generally  made  of  cast  iron,  which  makes 
a  neater  and  more  durable  piece  of  work. 

A  represents  the  lower  roller,  b  the  upper  roller,  c  D  a 
lever  which  presses  upon  the  brass  of  the  upper  roller,  e  f 
another  lever  to  increase  the  power  connected  with  cd. 
The  extremity  of  f  is  kept  down  by  a  pin  j  in  some  cases  a 
ht  is  used  in  place  of  the  pin. 


ESSAY    VIII. 


GEOMETRICAL  AND  PBACTICAL  METHODS 


lOK  rataaic  thb 


CENTRES  OF  GRAVITY  OF  MILL  WHEELS; 

ILLU8TRATXD  BT   BZAMPLBS, 

IN  WHICH  TWQ,  THREE,  AND  FOUR  WHEELS  COMPRISE  THE 
SYSTEM  UPON  ONE  AND  THE  SAME  SHAFT. 


463.  There  is  one  branch  of  mechanical  science  which 
belongs  essentiallj  to  mill-work  that  must  be  here  added 
to  these  Essays  of  Buchanan.  We  allude  to  Methods  of 
finding  the  Centre  of  Gravity  of  two  or  more  bodies  con- 
nected together  by  straight  inflexible  rods  passing  through 
their  respective  centres- 
Suppose  A  and  B  to  be  two  bodies  connected  together  by 

Fig.  1. 

Q o 

the  straight  inflexible  bar  a  b  passing  through  their  centres, 
and  it  were  required  to  find  the  centre  of  gravity  of  those 
bodies. 

At  the  points  a  and  b,  Fig.  2,  we  should  erect  the  per- 

FiG.  2. 


ESSAY  VIII.]      CENTRES  OF  GRAVITY  OF  MILL  WHEELS.     355 

pendiculars  a  c  and  b  d  of  any  convenient  length,  and  through 
c  draw  CD  parallel  to  ab  ;  then  we  should  produce  ac  to  f, 
Fig.  3,  and  make  c  e  to  ef  as  the  body  b  is  to  the  body  a.  Then 
joining  fd,  and  through  e  drawing  eh  parallel  to  fd,  and 

Fig.  3. 


firom  H  dropping  the  perpendicular  h  g,  the  point  g  would 
indicate  the  centre  of  gravity  of  the  two  bodies  a  and  b  ; 

for  A  :  B  ::  bg  :  ag 
hence  equating  the  products  of  the  extreme  and  mean 
terms 

A.  AG  3sB.bg 

From  which  we  infer,  that  when  two  bodies  connected  to- 
gether by  a  straight  inflexible  bar,  are  in  equilibrio,  the 
products  of  their  masses  multiplied  by  their  respective  dis- 
tanceSy  are  equal 

Let  a  =  the  mass  of  a 
b  ss  the  mass  of  b 
d  ss  the  distance  ag 
S  =  the  distance  bg 
and  D  =  the  distance  ab 
Then  according  to  the  foregoing  proportion, 

ad  ^  bSi  but  S  =  i}  ^  d 
consequently,  ad^hi}  -^  hd\  and 

d  =  J  ,  j> :  also  8  =  — —r- 


356  ON   THE   CENTRES   OF   GRAVITY  [[sSftAT  Vm. 

Consequently,  the  places  of  the  centre  of  gravity  is  known 
in  terms  of  the  masses,  and  the  distance  between  their  re- 
spective centres.     Hence  the  following  practical  role : 

464.  Multiph/  either  body  hy  the  whole  distance  be- 
tween their  centres  ;  divide  the  prodtLct  hy  the  sum  of 
the  bodies ;   the  quotient  will  be  the  distance  from  the 
centre  of  gravity  of  that  body  opposite  to  the  one  by  which 
the  whole  distance  is  multiplied. 

Example.  Let  the  two  bodies  be  respectively  4  and  7 
cwt. ;  and  their  distance  asunder  24  feet. 

Fio.  4. 

B  H  A 

G \ O 

7  * 

7  X  24 
Then  we  have  a  =  4,  J  =  7>  and  d  =  24,  or  -7 — —  = 

168 

-r-r  =  15^  feet,  being  the  distance  of  the  centre  of  gravity 

from  the  body  a. 

4  X  24       96 
Also  -^      ry'  =  11'^  ^^  ^*^^*'  being  the  distance  of  the 

centre  of  gravity  from  the  body  b. 

465.  The  example  supposes  the  connecting  rod  to  be 
void  of  weight ;  but  in  mechanics  this  is  never  the  case. 
The  same  law  must  obtain,  with  respect  to  the  portions  of 
the  connecting  rod,  that  we  saw  existing  in  the  mass  of 
each  body  multiplied  into  its  distance  from  the  common 
centre  of  gravity.  The  centre  of  gravity  of  an  imiform 
connecting  bar  must  be  at  the  middle  of  its  length  when 
that  bar  is  prismatic  or  cylindrical. 

If  ^  =  mass  or  weight  of  one  unit  or  length  of  the  bar, 

then  is  ^-5-  =  eflFective  energy  of  one  portion,  and  ^--r  = 

the  eflFective  energy  of  the  other;  and  these,  together  with 
the  eflFective  energies  of  the  bodies  a  and  b  referred  to  op- 


ESSAY  VIlI.j 

posite  sides  of  the  centre  of  gravity,  must  still  be  in  equi- 
librio ;  hence 

arf  +^  =  AS  +  ^ 
But  8  =  D  -  (^  and  therefore  by  substitution  we  obtain 
(a  +  fi  +  p-D)  rf  =  (6  +  t2-)  D, 

which  being  reduced  ^ves  the  following  equations 
rf_     (S&+;)d)d 
2  (a  +  6  +  p  rf) 

2(o  +  i  +  pd)' 
Hence  the  following  practical  rule  : 

466.  To  twice  the  weight  of  either  body,  add  the 
whole  weight  of  the  lever  or  connecting  bar,  and  multiply 
the  sum  by  tfie  central  distance ;  then  divide  the  pro- 
duct by  twice  the  mass  cmnpounded  of  the  bodies  and  tlie 
bar,  and  ths  quotient  will  be  the  distance  of  the  centre  of 
gravity  from  that  body  opposite  to  the  one  whose  double 
is  employed  in  the  first  step  of  the  operation. 

Example  1. — The  bar  is  24-  feet,  and  weighs  1  cwt., 
the  bodies  4  and  7  cwt.  respectively  as  before  ; 

Then  a=4;  b  =  T;  d  =  ^4-  feet,  and;)  s^jcwt.  .■.  d  = 
(a  X  7  +  Vt  X  g4)Q4  15  X  24  360 

2  (♦  +  7  +  A  X  24)  ~  2  (4  +  7  +  1)  ~  24  ~  ^^  ^^ 
being  the  distance  of  the  centre  of  gravity  from  a,  and 
therefore  24  —  15  =  9  feet,  the  distance  of  b  from  the 
centre  of  gravity;  for  9  +  15  =  24  feet. 

Example  2. — Let  the  bar  be  of  cast  iron,  42  feet  long 
and  252  lbs.  weight;  the  bodies  at  its  extremities  weighing 
13440  and  17920  Iba.  respectively. 

It  will  be  found  by  calculation  that  a  =  13440  and  b  = 
17920,  are  respectively  23-ifH  feet,  and  ISrlir  feet  from 
the  common  centre  of  gravity  of  the  bodies. 

467.  When  three  bodies  connected  together  by  a  straight 


S5S  an  the  csbtbbs  or  gkayitt       [essay  vui. 

mfleiible  bar,  are  in  eqoililHio^  die  product  of  one  body 
nmltqdied  hj  its  distance  from  the  ccmimon  centre  of  gra- 
vis of  the  STSleBy  is  equal  to  the  product  which  arises 
when  the  smn  of  the  other  two  bodies  is  multiplied  by  the 
distance  hetmeai  their  common  centre,  and  that  to  irhidi 
the  whole  system  is  referred*. 

If  a  =  mass  of  the  body  a  ; 
b  ==  mass  of  the  hoAj  b; 
l>  =  mass  of  |>  concentrated  in  |> ; 
d  =  distance  between  a  and  b  ; 
fi  =  distance  between  a  and/i, 
and  X  =  A  H  the  distance  between  a  and  the  common 

c^itre  H. 

Fio.  5. 

' 1 


Then  if  h  fiJls  between  a  and  />»  pH  :=  8  —  :r,  and  hb  = 
rf  —  x;  but  if  the  common  centre  falls  between  b  sndp, 
we  have  pn  =  or  —  S,  and  bh  ^  d  --  x\  and  in  either 
case  we  have 

(a  +  6  +  p)x  =^  bd  -h  p8 

(bd  +jpS) 


or  JT  = 


(a  +  6  +  jo) 

468.  The  practical  rule  is  the  following : 

Multiply  each  of  the  bodies  b  and  p  by  the  respective 
distances  from  a ;  then  divide  the  sum  of  the  products  hy 
the  aggregate  of  the  three  masses  for  the  distance  of  the 
centre  of  gravity  from  the  first  body  a,  to  which  the  dis- 
tance of  the  other  bodies  b  and  p  are  referred. 

Example. — ^Let  the  bodies  be  1 5,  20,  25  tons  respeo 
tively  ;  and  their  distance  12  and  16  feet  from  each  other; 
then  it  will  be  found  that 

ftc?  =  700;  pS  =  240;  mdbd  -{-pS  =  940 

*  Dr.  Jaoiiesons  Mechanics  for  Practical  Men.     Loodoiiy  1837,  Svo. 


BSAT  viir.3 


OF    MILL    WHEELS. 


359 


but  (a  +  b  +  p)  =  GO;  therefore  z  is  distant  from  a  by 

-qq   =  15f  feet ;  x  —  S  =  3f  =  the  distance  of  j:  fromp  ; 

1  d  —  j:  =  12^  feet  =  distance  from  b  :  that  is  to  say  ; 
AH  =  15§  feet,  or  the  distance  of  a  from  h 
pu  =    3f  feet,  or  the  distance  o( p  firom  h 
and  BH  =  12J  feet,  or  the  distance  of  b  from  h. 
Hence  ah  +  hb  =  15|  +  12^  =  28  =  l6  +  12  feet. 
469-  These  results  Dr.  Jamieson  verifies  by  the  follow- 
ing construction  in  his  "Mechanics^  Practical  Men."' 

Fia.  6, 


Draw  the  straight  line  ab,  and  from  a  scale  of  equal 
"parts  make  Ap  =  12  and  ps  =  l6  feet  j  through  the  point 
B  draw  the  straight  line  bf  in  any  direction  with  respect  to 
AB  ;  make  be  =  20,  and  ef  =  25,  the  numbers  which  re- 
spectively express  the  magnitudes  of  the  bodies  ^>  and  b 
acting  on  the  straight  line  a  b,  at  the  points  p  and  b  ;  join 
rp,  and  through  the  point  e  draw  eg  parallel  to  Fp,  which 
produce  to  c,  and  makcGD  =  15,  the  number  which  cx- 
iresses  the  magnitude  of  the  body  a  acting  at  a,  and  make 
be  =  45,  the  number  =  sum  of  p  and  b  acting  at  g  :  join 

*  Article,  Centre  of  Gmvitj,  pp.  IS,  20. 


360  ON   THE   CENTRES   OF   GRAVITT         []eSSAT  VHI. 

c A,  and  through  d  draw  dh  parallel  to  c  a;  theA  is  h  the 
place  of  the  centre  of  gravity  of  the  three  forces  Oy  p^  and 
hy  acting  at  the  points  a,  p^  and  b  of  the  har  ab  ;  and  ah, 
pay  and  b  h,  their  respective  distances,  which  if  measured 
from  the  scale  will  be  found  equal  to  15f,  3f,  and  IS^ 
feet  respectively. 

Workmen  may  be  informed,  that  in  constructions  of  this 
kind  it  is  not  necessary  to  take  the  numbers  which  express 
the  magnitudes  of  the  bodies  from  the  same  scale  as  those 
which  express  their  relative  distances  ;  for  since  they  are 
magnitudes  dissimilar  to  one  another,  they  cannot  be  com- 
pared ;  consequently  the  ratio  or  proportion  of  the  numbers 
is  all  that  we  require :  all  the  magnitudes  of  the  same  kind 
must  however  be  taken  from  the  same  scale. 

This  remark  is  made  because  some  of  the  foregoing 
numbers  express  weight,  others  lineal  measure ;  in  setting 
off  their  relations  we  used  the  same  scale  for  all ;  but  this 
is  not  necessary. 

470.  The  cases  of  utility  consistent  with  this  theorem 
are  only  three ;  viz. 

1.  When  p  is  less  than  a  +  6,  but  such  that  a  -h  j»  is 
greater  than  6,  and  h  +  p  greater  than  a  j 

2.  When  p  is  equal  to  a  +  6 ; 

3.  When  p  is  greater  than  a  +  J. 

The  equation  of  equilibrium  is  the  following,  which  we 
borrow  from  the  "  Mechanics  for  Practical  Men." 

d 

The  following  examples  are  given  to  show  persons  un- 
acquainted  with  algebra  how  they  may  apply  the  principle 
now  before  them. 

Example  1. — At  the  extremities  of  an  iron  shaft  22 
feet  long  are  fixed  two  wheels,  a  and  6,  respectively  2  and 
2^  cwt. ;  and  somewhere  between  these  another  wheel,  />, 
is  fixed,  1^  cwt ;  at  what  distance  from  each  of  the  ex- 


I  ESSAY  VIIJ.3 


OF   MILL    WHEELS. 


361 


treme  wheels  must  the  intermediate  one  be  fixed,  so  tliat 
the  whole  weight  may  come  upon  the  middle  of  the  shaft, 
I  when  it  is  supported  by  a  transverse  bearer  h  ? 


I 


Here  a  =  Q;  p  =  1'5;  b  =  2*5,  and  rf  =  22  teat,  sup- 
posed to  be  the  distance  between  the  centres  of  the  extreme 
wheels  ;  then  since  the  shaft  is  supported  on  its  gudgeons 
■at  the  extremities,  and  on  the  journal  at  the  transverse 
bearer,  we  may  consider  it  as  having  no  effect  upon  the  sys- 
tem of  wheels  as  regards  the  place  of  the  centre  of  gravity ; 
therefore  by  substituting  the  above  numbers  in  the  fore- 
going equation,  we  have 

^^  (2  +  1-5  ~  2-5}  =  ??  X  1  =  74  feet : 
X  1-5^  \      3  '3         ' 

being  the  distance  of  the  lighter  wheel  a  from  p ;  but  the 
middle  of  the  shaft  is  1 1  feet  from  a  or  i ;  therefore  we 
have  11  —  7^  =  3|  for  the  distance  of;*  from  the  journal ; 
id  11  +  3|  =  Hf  for  its  distance  from  the  greater 
'heel  b, 
4'71.  When  the  distance  is  known  or  limited  by  situa- 
tion, as  in  practice  frequently  happens  to  be  the  case,  the 


gpB 


'Here  wo  have  given,  as  is  plain  from  the  terms  on  the  right 


I  equation  becomes 
Here  wo  have  give: 


36S 


ON -THE  CBNTRES  OF  OEATITT 


[» 


hand  ride  of  the  equation,  the  magnitudee  of  the  three 
bodies  a,  p  and  h,  acting  in  the  same  straight  line,  and  the 
distance  between  the  middle  body  p,  and  the  first  extreme 
a ;  and  we  are  required  to  find  d,  the  distance  between  the 
extreme  bodies  a  and  b,  and  that  the  common  centre  of 
gravity,  or  the  centre  of  the  system  shall  fall  at  the  middle 
of  that  distance. 

473.  Suppose  then  for  illostration  of  the  case  we  take 
the  following 


Example. — The  shaft  of  a  mill-wheel  has  to  sustain 
three  wheels  of  the  weights  of  %  11,  and  3  cwts. ;  what  must 
be  the  length  of  the  shaft  from  centre  to  centre  of  the  ex- 
treme wheels,  in  order  that  a  transverse  girder  placed  at 
the  middle  of  its  length  shall  release  the  gudgeons  from 
the  pressure  and  sustain  the  system  at  rest,  the  distance 
between  the  first  extreme  and  the  intermediate  wheels 
being  12  feeL 

Here  we  have  ^ven  by  the  question,  a  =  2,  f>  =  7,  i  = 
3,  and  S  =  12;  then  writing  these  numbers  for  their  con- 
stituents in  the  equation,  we  have 

2  X  7  X  12       168 


rf  =  , 


=  28  feet. 


1+7-    3 " 
for  the  distance  between  centre  and  centre  of  the  extreme 


KS8AT  VIII.j 


OF    MILL    WRBBLS. 


363 


=  4-8  feet. 


wheels  a  and  h ;  consequently  the  place  we  must  assign  as 
the  common  centre  of  gravity  is  It  feet  from  either  ex- 
:^me  end,  and  2  feet  from  the  place  of  p  the  intermediate 
|rheeL 

I  473.  To  verify  this  result,  we  may  compute  the  place  of 
>  common  centre  of  gravity  of  the  two  wheels  p  and  h  by 
iie  first  problem,  in  which  case  we  have 
16x3 
7  +  3  ' 
s  the  distance  from  p,  consequently  the  distance  between 
the  centre  of  the  system  and  that  of  the  two  bodies  p  and  b 
is  4*8  —  2  =  2*8  feet ;    then   reasoning  by  the   previous 

tustration,  we  have 
14a  =  2-8  (p  +  b),  that  is 
14  X  2  =  2-8  X  10  =  28,  as  before. 
r,  to  numerous  machinists  who  are  masters  of  algebra,  if 
I  put  X  =  the  distance  of  each  extreme  wheel  from  the 
centre  of  the  shaft,  then  2  j:  =  the  whole  length  of  the 
shaft,  and  jr  —  12  =  the  distance    of   the   intermediate 

Iieel ;  hence 
2x  +  (_x  -IQ)  x7  =  3x 
or  7.r  -  84  =  X  ;  1.  e.  6  .r  =  84  ; 
jrefore  x  =  14,  and  14  x  2  =  28  as  before. 
474.  When  the  weight  of  the  axle  of  the  wheels  is  given, 
we  may  adopt  7c  to  express  that  element.    Then,  if  the  bar 
be  of  uniform  shape  and  density,  the  centre  of  gravity  of 
each  segment  made  by  the  centre  of  the  system,  will  occur 
at  the  middle  of  its  length,  and  the  weight  of  the  segment 
^^riil  be  expressed  by  wx,  and  iv  (d  —  x)  respectively.     The 
^BpTective  strength  of  the  energies  is  then 
^f  4  tci-*,  and  ^  w  (d—x)\ 

consequently  in  the  case  of  an  equilibrium,  we  shall  have 
ax  -\-  p  (J'-S)  +  ^wx^  =  b  (rf-.r)  +  ^  w  (d-x)*, 
+  4  M^r"  =  b  (d-x)  +  p  (S-.r)  +  ^w  (rf-.r)*; 


364 


ON    THE    CENTRES   OF   GRAVITY.  [eSSAY  IU. 


but  in  either  case,  when  the  equations  are  properly  re- 
duced, we  6nd  generally  that 

^_(g&  +  w  d)d   +   gpg 
2(a  +  i+j>  +  «!(/)' 
The  following  practical  example  will  bring  this  compli- 
cated equation  into  a  readable  form,  better  than  could  be 
1  by  a  rule. 

FiQ.  9. 


Example. — A  cast  iron  shaft,  4  inches  square,  and  3Q  feet 
long  between  gudgeon  and  gudgeon,  is  required  to  sustain 
three  wheels,  whose  weights  are  4,  7,  and  6  cwt.  respect- 
ively, placed  at  the  distance  of  14  and  33  feet  from  each 
other.  At  what  point  of  the  shait  must  an  iipright  be 
placed  to  remove  the  pressure  entirely  from  the  gudgeon^ 
and  balance  the  shaft  with  all  its  apparatus. 

Here  we  have  given  the  wheels  a  =  4,  j)  n  7,  and  ft  b 
6,  and  the  shaft  d  =  36,  also  S  =  14.  Writii^  then 
numbers  for  their  correspondent  symbols  in  the  finregonig 
equation,  we  shall  have 

(g  X  6  +  36  «?)  36  +  g  X  7  X  14 
^'  2  (4  +  6+7+36  w)  ' 

Now  since  the  material  of  which  the  shaft  it  nuidB  3| 
cast  iron,  the  weight  of  1  foot  in  length,  or  the  value  of  w 
is  4   X   4    X  3'2  =  dl*3lb6.*;  conseqaently,  by  substi- 

*  THe  wd^t  of  a  bar  of  cast  iron  one  inidi  aqona  and  1 2  inches  loug, 
ia  3-S  Iba.,  the  mnldpUer  nnd  m  the  qoeitton. 


ESSAY  VIll.]  OF    MILL    WHEELS. 

toting  51-2  instead  of  w  in  the  foregoing  value  of  x,  we 
shall  obtain 


.(^ 


36 


-51-2)  X  36  +  g  X  7  . 


2(4  +  6  +  7+  36    X    51-«) 
18  feet  very  nearly. 

Therefore,  the  place  of  the  support  is  at  18  feet  from 

each  of  the  extreme  wheels,  and  4  feet  from  the  inter- 

I'lnediate  one ;  but  if  the  weight  of  the  shaft  had  not  been 

ken  into  the  estimate,  we  should  have  had  x  =  18  -j^- 

hence  the  effect  which  this  element  produces  is,  to 

ice  the  support  ^  of  a  foot,  or  very  nearly  6  inches  more 

B  way  than  the  other,  a  quantity  which  in  large  construc- 

ttiB  may  be  disregarded.     But  it  was  necessary  to  shew 

gtiiat  we  ahould  not  consider  the  axle  void  of  weight  in  our 

*  calculations,  especially  where  their  accuracy  may  be  tested 

by  other  persons  who  would  not  allow  this  element  to  be 

thrown  out  of  the  equation  of  equilibrium. 

H    Of  the  centre  of  gravity  of  Jour  or  more  bodies  situated 
H  in  the  same  right  line. 

^^     475.  This  is  but  an  extension  of  the  previous  case :  in- 
^^eed  the  law  of  continuation  is  so  obvious,  that  we  shall 
make  one  example  suffice  for  its  illustration  ;  but  to  make 
the  way  smooth,  let  a,  p,  n,  b,  represent  the  four  bodies 
I  taken  in  order,  from  a  the  first,  to  b  the  last. 


¥ 


i-+-i F 


Let  i  denote  the  distance  from  «  to  7^,  and  S^the  dis- 
tance from  «  to  n,  and  d  the  distance  from  a  to  i.     Also 
X  denote  the  distance  from  a  to  the  place  of  the  com. 

centre  of  the  whole  mass. 
If  then  the  bodies  a  and  p  are  on  one  side  of  the  com- 
mon centre,  while  the  other  two  bodies  »  and  b  are  situate 
the  other  side  of  that  centre,  we  shall  have  x ;  (.r  —  t)  ; 


tanci 
^niion 


_4ai  the  other  side  01  t 


366  ON   THE   CENTRES   OF   GRAVITY        |^£88AT  TID. 

(fi'  —  or),  and  (^d  —  x)  for  the  respectiye  distances  of  Ad 
bodies  from  the  centre  of  gravity ;  consequently,  by  the 
principle  ah*eady  indicated,  we  have 

a  ^  +  p  (^  —  S)  =  n  (8^  —  or)  -f  6  (rf  —  x) ; 
which  by  transposition  and  division  gives 

^a  +  p  +  n+6'' 
And  we  may  write  this  equation  thus,  for  the  benefit  of 
such  readers   as  may  require  its  meaning   in  words  at 
length. 

476.  Rule. — Multiply  the  magnitude  or  density  of  each 
body  by  its  respective  distance  from  the  beginning  of  the 
system^  and  divide  the  sum  of  the  products  by  the  sum  of 
the  bodies  for  the  distance  of  the  centre  of  gravity  sought. 

Example. — Four  bodies  connected  by  a  straight  inflex- 
ible bar,  have  their  weights  respectively,  18,  26,  12,  and 
30  cwt. ;  and  their  distances  from  each  other  are. 

From  a  to  |7,  I7  feet, 
a  to  71,  23  ditto, 
a  to  i,  40  ditto. 

At  what  point  in  the  length  of  the  bar  shall  the  common 
centre  of  gravity  be  marked  ? 
Here  we  have  given 

a  =  18;  71  =  12;  S  =  17;        ,   ^ 

^  =  26;   6  =  30;   S'  =  23 ;  ^^^  ^  =  ^' 

Let  these  values  of  the  elements  of  the  system  be  sub- 
stituted  in  lieu  of  the  symbols  in  the  foregoing  equation, 
and  it  reads 

r  =  ^^   X   17  +   12   X  23  -h  30  X  40  _  ^^  ^ 

18+26  +  12+3  ^"^ 

feet  from  a\  5*3  feet  from  p ;  iV  from  n ;  and  I77 
from  b. 

477.  We  shall  now  exhibit  the  principle  of  continuation 


BSSAT  Vni.3 


OF   MILL   WHEELS. 


S67 


by  a  geometrical  construction,  in  which  the  reader  may 
trace  with  great  facility  the  combinations  involved  in  the 
equation  we  have  just  worked  out  for  him. 

Let  A  B  be  a  straight  line  passing  through  the  centres  of 
the  four  bodies,  a,  p,  n,  b. 

Fig.  10. 


Make  a  b  =  40  feet,  taken  from  a  scale  of  equal  parts. 
On  the  straight  line  a  b  set  off  Ap  and  a  n  equal  respect- 
ively to  17  and  23  feet,  taken  from  the  same  scale  as  ab. 
Then  are  the  points  a,  p,  n,  and  b,  the  positions  of  the 
four  bodies,  the  weights  of  which  constitute  the  elements 
of  the  question,  and  whose  common  centre  of  gravity  we 
shall  now  trace  by  completing  the  construction  of  the  dia- 
gram. 

Through  the  point  b  draw  the  straight  line  b  f  in  any 
direction  at  pleasure ;  make  b  e  proportional  to  the  weight 
of  the  body  n,  and  ef  to  that  of  the  body  b ;  join  fti  ;  and 
through  the  point  e  draw  eg  parallel  to  rn ;  then  is  the 
point  G  the  common  centre  of  gravity  of  the  bodies  b 
and  71. 

Next  produce  e  g  to  c,  making  g  d  proportional  to  the 
weight  of  the  body  p,  and  do  =  bf,  or  proportional  to 
the  sum  of  the  bodies  6  and  n  i  join  cpf  and  through  the 

bb2 


368  ON    THE    CENTBES   OF   GRAVITY        [bSSAT  TQI. 

pdnt  D,  draw  oh  parallel  to  c^.  Then  is  the  prant  a  the 
common  centre  of  gravity  of  the  three  hodies,  p,  n,  b. 

Finally,  produce  dh  to  the  point  e,  making  hi  pn^- 
tional  to  the  weight  of  the  body  a,  and  ik  equal  to  go,  cr 
proportional  to  the  sum  of  the  bodies  p^  n,  and  b ;  ysa 
KA,  and  through  the  point  i,  draw  ix  parallel  to  ka  \  thai 
shaU  the  point  x  on  the  line  ab  be  the  common  centre 
of  gravity  sought'. 

For  we  have  bf  =  n   +   6-;  an  =  d  —  fij  ep  =  &} 

,  i  (rf  -  8)  +  B  (y  -  8).  ^, 

n  +  6  * 

_  hd+pt  +  ng. 
n  +  ;»  +  6  ' 

Ki  =p  +  n  +  6;  and  hx  =  \ t-E — '*"  "1  an  eqiia- 

(  o  +;»  +  n  +  b  i 

tion  which  is  identical  with  that  from  which  we  demosD- 
strated  the  example,  and  which  we  shall  now  turn  to  ac- 
count in  the  solution  of  another  bearing  immediately  on 
the  subject  of  this  essay,  and  with  which  also  it  is  our  in- 
tentioD  to  bring  it  to  a  close. 


p  +  M  +  6 ;  g;>  = 

+  P  +  n  +  6; 


as  n  +  i:  HK 


Example. — Four  cast  iron  wheels,  the  weights  of  which 
are  respectively  -t,  5,  6,  and  7  cwts.,  are  fixed  upon  a  shaft 
at  the  scraral  distances  of  8,  10,  and  12  feet  apart,  taken 

*  In  Dr.  Jainiesoii's  "  Mecluuiin  for  Practic*]  Men,'  there  ia  ta  elc«u>t 
dratoiutntian  of  thii  coattnictiou. 


ESSAY  VIII.]  OF   MILL   WHEELS.  869 

in  order ;  then,  if  the  shaft  have  no  influence  upon  its  po- 
sition,  what  is  the  distance  of  each  wheel  from  the  common 
centre  of  gravity  of  the  system  ? 

There  are  here  supplied  by  the  question,  seven  terms  of 
its  elements,  and  the  eighth  is  to  be  found  thus,  agreeably 
to  the  foregoing  equation ;  viz., 

a  =  4;jE>  =  5;n  =  6;6  =  7;S  =  8;8'  =  18; 
and  d  ^  SO;  hence,  if  we  substitute  these  numerical  values 
for  their  corresponding  symbols,  we  shall  have 

^      5x8+6x18+7x80       ,^, 

X  =     =  lO  Ti 

4  +  5+7 
feet  from  the  body  a;  8i^  fi^m  p^  li\-  from  n;   and 
18A  fi^ni  6. 

The  numerical  operation  deducible  from  the  geometrical 
construction,  furnishes  three  elegant  proportions.  For  by 
gunilar  triangles,  beg  and  BPn,  we  have 

BF  :  Bn  :  :  EF  :  no ;  that  is 
(n  +  b)  :  (d  -  O  :  :  6  :  no  =  *IfLl^;  which 

arithmetically  becomes 

6  +  7  :  80  -  18  :  :  7  :  MG  =  ?|  =  6-|. 

Again,  in  the  similar  triangles  hdg  and pcG,  we  have 
GC  :  Gj)  :  :  DC  :  pH ;  that  is 

(p  +  n  +  b)  :  — i^ ^ r^^ -^  :  :  »  +  ft  :  oh 

^^  ^  n  +  ft  ^ 

p  -h  n  -{•  b 
arithmetically  is  written 

5  +  6  +  7  :  ^  +  (18  -  8)  :  :  6  +  7  :i>H  =^  =  ll| 

Finally,  in  the  similar  triangles  hio;  and  hka,  we  have 
the  following  proportion, 


370  ON   THE   CENTRES   OF   ORAVITT        f  ESSAY  Vm. 

HK  :  HA  :  :  Ki  :  Ax;  that  is, 

(a  +  p  +  n  +  6  :  fifL±-£l±^l :  :  («  +  n  +  6)  :  a j; 

I    />  +  »  +  6     J 

which  by  equating  the  product  of  the  extremes  and  means 
gives  the  elegant  equation  that  preceded  the  example,  and 
which  arithmetically  is  written  out  thus : 

(4  +  5  +  6  +  7)  :  (^  +  8)  :  :  (5  +  6  +  7)  :  ax  =  ^; 

consequently  by  division  the  fraction  becomes  ax  =  16^ 
feet,  the  same  as  before. 

478.  Thus  we  have  traced  fit)m  principles  of  the  greatest 
sunplicity  the  theory  of  the  common  centre  of  gravity  for 
so  much  of  null- work  as  has  reference  to  systems  of  wheels 
arming  the  same  axle.  We  shall  close  this  article  by  re- 
marking in  reference  to  bevel  gear,  that 

1.  The  centre  of  gravity  of  the  surface  of  a  cone  is  the 
same  as  the  centre  of  gravity  of  its  triangular  section. 
And  the  centre  of  gravity  of  a  right  cone  is  situated  at 
f  ths  of  the  axis  from  the  vertex :  or  ^th  from  the  base  of 
the  cone.  The  same  holds  good  of  any  pyramid  whose 
base  is  a  polygon. 

2.  The  centre  of  gravity  of  the  surface  of  a  conic  frus- 
tum is  the  same  as  the  centre  of  gravity  of  the  trapezoid 
formed  by  a  plane  passing  along  the  axis.  And  the  centre 
of  gravity  of  the  conic  frustum,  when  its  height  and  the 
diameters  of  the  two  ends  are  given,  is  detemuned  by  the 
following  equation, 

Where   h  is    the  height  of  the  conic  frustum ;  r  is  the 
radius  of  the  less  end ;  r  the  radius  of  the  greater ;  and  S 
represents  the  distance  of  the  centre  from  the  less  end. 
The  practical  rule  is  this : 


I  ESSAY  Vltl.] 


OF    MILL    WHEELS. 


371 


Tq  the  gum  of  the  sfjxiares  of  the  rndii  of  the  two  ends 
add  their  product,  then  multiply  the  sum  by  4,  aiid  reserve 
the  result  for  a  divisor. 

To  three  times  the  square  of  the  radius  of  the  greater 
,  add  the  square  of  the  radius  of  the  less  end,  toge- 
•  with  twice  tfie  product  of  the  radii,    and  multiply 
%^  sum  hy  the  height  of  the  frustum  for  a  diimlend. 

Tlien,  divide  the  dividend  by  the  reserved  divisor,  and 
the  quotient  will  express  the  distance  between  the  centre  of 
magnitude  of  the  less  end,  nnd  tlie  centre  of  gravity  qftlie 
istwn. 

S.  The  centre  of  granty  of  the  surface  of  a  cylinder  is 
the  same  as  the  centre  of  gravity  of  the  parallelograni 
made  hy  the  plane  passing  through  the  axis. 
_  4.  The  distance  of  the  centre  of  gravity  of  a  circular 
^birc  from  the  centre  of  the  circle  is  a  fourth  proportional  to 
^nhe  length  of  the  are,  the  radius  of  the  circle,  and  the 
Htdiord  of  the  arc. 

^1  5.  The  ordinate  of  a  common  parabola  is  a  mean  pro- 
portional between  the  abscissa  and  the  parameter  of  the 
axis ;  and  the  position  of  the  centre  of  gravity  is  in  the 
axis  of  the  figure,  and  at  the  distance  of  three  fifths  of  the 
jscisaa  from  the  vertex. 

y  6.  The  centre  of  gravity  of  any  semiparabola  occurs  in 
ke  ordinate  of  the  axis,  passing  through  the  centre  of  gra- 
Hty  of  the  whole  parabola, 

7.  The  distance  between  the  vertex  and  the  centre  of 
gravity  of  a  parabolic  conoid,  is  equal  to  two  thirds  of  the 
axis. 
^B    There  are  several  other  figures  that  occur  in  mill-work, 
|Bint  the  discussion  we  have  entered  into  has  spun  out  much 
^twyond  the  limit  we  had  assigned  to  it,  and  we  are  there- 
fore compelled  to  refer  the  reader  to  other  treatises,  which 
Ipter  into  the  composition,  revolution,  and  properties  of 


372      CENTRES  OF  ORAVITY  OF  MILL  WHEELS.     [»S8AT  YUI. 

bodies  in  motioiiy  for  all  such  matter  as  should  be  known  to 
complete  the  education  of  a  sound  millwright. 

479-  The  tables  of  squares  and  cubes  which  are  annexed 
will  be  acceptable,  as  also  the  square  roots  and  cube  roots 
of  all  numbers  from  1  to  1000,  which  have  been  taken 
from  Hutton's  **  Course  of  Mathematics,''  and  will  be  found 
very  useful  on  many  occasions. 


■ 

■ 

373 

Squ»e. 

Cube. 

SquueRoot. 

CubeRooL 

1 

1 

1 

I -0000000 

1-000000 

2 

4 

8 

1-4 14-2 136 

1-259921 

3 

» 

27 

1-7320508 

1-442250 

4 

10 

61 

20000000 

1-687401 

& 

25 

125 

2-2360680 

1-709976 

0 

36 

216 

2 '4404897 

i81712l 

7 

49 

343 

2"  64575 13 

1-012931 

8 

64 

512 

2-8284271 

2-000000 

S 

81 

729 

3'0000000 

2-080084 

10 

100 

1000 

31622777 

2-154435 

It 

121 

1331 

3-3166248 

2-223080 

12 

144 

172)1 

3-4641016 

2-289428 

13 

169 

2197 

36065513 

2-36 1336 

14 

196 

2744 

3-7416574 

2-410142 

16 

225 

3375 

36729833 

2-466212 

16 

256 

4096 

4-0000000 

2-619842 

17 

289 

4913 

41231050 

2-571282 

16 

324 

6832 

4-2426407 

2-620741 

19 

361 

6659 

4-3688989 

2-668402 

20 

400 

8000 

4-4721360 

2-714418 

■ 

21 

441 

B2BI 

4-682.5757 

2-738923 

■ 

22 

484 

10(M» 

4-6004158 

2-802039 

■ 

23 

629 

12167 

4-7958315 

2843867 

34 

576 

13824 

4-8989705 

2-884490 

2S 

625 

15625 

50000000 

2-024018 

26 

676 

17576 

5-09B0195 

2-962496 

' 

27 

729 

19683 

5-1961624 

3-000000 

n 

28 

784 

21952 

5-2915026 

3-036580 

■ 

29 

841 

24389 

6-3851648 

3-072317 

■ 

30 

900 

27000 

6-4772256 

3-107232 

■ 

31 

061 

29791 

5-6677644 

3-141381 

■ 

32 

1024 

32768 

6-6568642 

3174802 

■ 

33 

1089 

33037 

6-7445626 

3-207534 

■ 

34 

1156 

39304 

6-8309519 

3-239612 

n 

35 

1225 

42875 

5-9160798 

3-271066 

36 

1296 

46656 

6-0000000 

3-301927 

37 

1369 

50653 

60827625 

3-332222 

1 

38 

1444 

54872 

6-1614140 

3-361975 

^ 

39 

1521 

69319 

6-2449980 

3-391211 

■ 

40 

1600 

64000 

6-3245&53 

3-419062 

■ 

41 

1681 

08021 

6-4031242 

3-448217 

■ 

43 

1764 

74088 

6-4807407 

3-476027 

■ 

43 

1840 

70507 

6-55743a5 

3-503398 

■ 

44 

1936 

85184 

6-6332490 

3-530348 

f 

4S 

2026 

81 125 

6-7082039 

3-55<i803 

46 

2116 

07336 

6-7823300 

3-583048 

47 

2209 

103823 

6-8656546 

3-608826 

J 

48 

2304 

1106(t2 

6-92H2032 

3-634241 

40 

2401 

117649 

7-0000000 

3-669306 

H 

^^ 

60 

2600 

126000 

70710678 

3-684031 

H 

■ 

51 

2601 

132651 

71414284 

3-708430 

1 

I 

1 

^ 

974 


NuBbar* 

Si|iiare. 

Cube. 

fl^joue  Root. 

Cube  Root 

63 

2704 

140608 

7*2111026 

3-732611 

68 

2809 

148877 

7-2801099 

3-756286 

64 

2916 

167464 

7-3484692 

3-779763 

66 

3026 

166376 

7-4161966 

3-802053 

66 

3136 

176616 

7-4833148 

3-825862 

67 

3249 

186193 

7-6498344 

3-848501 

68 

3364 

196112 

7-6167731 

3-870877 

60 

3481 

206379 

7-6811467 

3-802996 

60 

3600 

216000 

7-7469667 

3-014867 

61 

3721 

226961 

7-8102497 

3-036407 

62 

3844 

238328 

7-8740079 

3-057802 

63 

3969 

260047 

7-9372639 

3-070067 

64 

4096 

262144 

8-0000000 

4-000000 

66 

4226 

274625 

8-0622677 

4-020726 

66 

4356 

287496 

8-1240384 

4-041240 

67 

4489 

300763 

8-1853528 

4-061648 

68 

4624 

314432 

8-2462113 

4-081666 

69 

4761 

328509 

8-3066230 

4-101566 

70 

4900 

343000 

8*3666003 

4-121286 

71 

6041 

367911 

8-4261498 

4-140818 

72 

6184 

373248 

8-4852814 

4-160168 

73 

6329 

389017 

8-6440037 

4-170330 

74 

6476 

406224 

8-6023263 

4-106336 

76 

6625 

421876 

8-6602640 

4-217163 

76 

6776 

438976 

8-7177979 

4-236824 

77 

6929 

456533 

8-7749644 

4-254321 

78 

6084 

474552 

8-8317609 

4-272659 

79 

6241 

403039 

8*8881944 

4-290841 

80 

6400 

512000 

8-944*2719 

4-308870 

81 

6561 

531441 

9-0000000 

4-326749 

82 

6724 

551368 

90553851 

4-344481 

83 

6889 

671787 

9-1104336 

4-362071 

84 

7056 

592704 

9-1651514 

4-379519 

85 

7225 

614125 

9*2195445 

4-396830 

86 

7396 

636056 

9-2736185 

4-414005 

87 

7569 

658503 

9-3273791 

4-431047 

88 

7744 

681472 

9*3808315 

4-447960 

89 

7921 

704969 

9-4339811 

4-464745 

90 

8100 

729000 

9-4868330 

4-481405 

91 

8281 

753571 

9-5393920 

4-497942 

92 

8464 

778688 

9-5916630 

4-514357 

93' 

8649 

804357 

9-6436508 

4-530655 

94 

8836 

830584 

9-6953597 

4-546836 

95 

9025 

857375 

9-7467943 

4-562903 

96 

9216 

884736 

9-7979590 

4-578857 

97 

9409 

912673 

9-8488578 

4-594701 

98 

9604 

941192 

9-8994949 

4-610436 

99 

9801 

970299 

9-9498744 

4-626065 

100 

10000 

1000000 

lOOOOOOOO 

4-641589 

101 

10201 

1030301 

10-0498756 

4-657010 

102 

10404 

1061208 

10-0995049 

4-672330 

375 

^ 

^ 

^ 

Nmnlxr. 

8qu«e. 

Cube. 

Sqiuro  Root. 

Cube  Root 

I 

103 

10609 

1092727 

101488916 

4-687548 

104 

10810 

H248G4 

10-1980390 

4-702669 

^^^H 

105 

11025 

1 157625 

10-2169508 

4-717694 

^^^H 

100 

11230 

IIBIOIO 

10-2966301 

4-732624 

^^^H 

107 

11449 

1226043 

10-3440804 

4-747459 

^^^H 

108 

11001 

1269712 

10-3923046 

4-702203 

^^^H 

109 

11881 

1295029 

10-4403065 

4-776856 

^^^H 

110 

12100 

1331000 

10-4880885 

4-791420 

^^^H 

111 

123-21 

1367631 

10-5366538 

4805896 

^^^H 

112 

12544 

1404928 

10-6830052 

4-820284 

113 

12709 

1442897 

10-6301458 

4-834588 

^^^H 

114 

12096 

1481544 

10-6770783 

4-848808 

^^^H 

116 

13223 

1520875 

10-7238053 

4-862944 

^^^H 

11« 

13496 

1560896 

10.7703296 

4-876999 

^^^H 

117 

1368» 

1601613 

10-8166538 

4-890973 

^^^^1 

118 

13924 

1043032 

10-8627805 

4-904868 

^^^^1 

119 

14101 

1686169 

10-Wlb7r21 

4-918685 

^^^^H 

130 

14400 

1728000 

10-9644512 

4-932424 

^^^^H 

121 

14641 

1771661 

11-0000000 

4-946088 

^^^^H 

122 

14884 

1815848 

11-0453610 

4-9.S9675 

^^^^H 

123 

15129 

1860867 

11-0905365 

4-973190 

^^^^1 

124 

15376 

1006624 

11-1355287 

4-986631 

125 

15625 

1953125 

11-1803399 

5000000 

^^^^1 

120 

15876 

2000376 

11-2249722 

5-013298 

^^^H 

127 

16129 

2048383 

11-2694277 

5-026526 

128 

16384 

2097152 

11-3137085 

5-039684 

^^^^1 

129 

16041 

2146089 

11-3578167 

5052774 

^^^^1 

180 

18900 

2197000 

11-4017543 

6-065797 

^^^^1 

131 

17161 

2248001 

11-4455231 

6-078753 

^^^H 

132 

17424 

2299968 

11-4891253 

6-091643 

^^^H 

133 

I76B9 

2352637 

11-5325626 

6-104469 

^^^H 

184 

17956 

2406104 

11-5758369 

5-117230 

^^^H 

13fi 

18225 

2460375 

11-6189500 

5-129928 

186 

18496 

2516456 

11-6619038 

5-142563 

^^^H 

137 

18769 

2571353 

11-7046999 

5-155137 

^^^H 

13U 

19044 

2628072 

11-7473444 

6-167649 

^^^H 

139 

19321 

2685619 

11-7898261 

5-180101 

^^^H 

140 

19000 

2744000 

11-8321596 

5-192494 

^^^H 

141 

19881 

2803221 

11-8743421 

5-204828 

^^^H 

142 

20]  04 

2803288 

11-9163753 

5-217103 

^^^H 

143 

20449 

2B24207 

6-220321 

^^^H 

144 

20730 

2085084 

12000<t000 

6-24 1482 

^^^H 

145 

21025 

3048625 

12-0415946 

5-253588 

148 

21310 

3112136 

12-0830460 

6-265637 

^^^H 

147 

21609 

3176523 

12-1243557 

5-277632 

^^^^1 

148 

21904 

3241792 

12-1055251 

6-289572 

^^^^M 

14& 

22201 

3307919 

12'2065656 

5  301469 

^^^^M 

150 

22500 

3375000 

12-2474487 

6-3 13-293 

^^^^M 

151 

22801 

3442951 

12-2882057 

6-325074 

^^^^H 

152 

23104 

3511808 

1 2-3288280 

6-336803 

^^^^H 

153 

23409 

3581677 

12-3693169 

6-348481 

^^1 

^^^H^^^fl 

376 


NumMT* 

Cube. 

SqaneRoot 

Cube  Root 

164 

23716 

3662264 

12-4096736 

5-360108 

156 

24026 

3723876 

12-4498996 

5-371685 

156 

24336 

3796416 

12-4899960 

5-383213 

157 

24649 

3869893 

12-5299641 

5-394690 

158 

24964 

3944312 

12-5698061 

5-406120 

159 

26281 

4019679 

12-6095202 

5*417501 

160 

25600 

4096000 

12-6491106 

5-428835 

161 

25921 

4173281 

12-6885775 

5-440122 

162 

26244 

4251528 

12-7279221 

5-461362 

168 

26569 

4330747 

12-7671458 

5-462656 

164 

M896 

4410944 

12-8062485 

5-478703 

165 

27225 

4492125 

12-8452326 

5*484806 

166 

87556 

4574296 

12-8840987 

5*495866 

167 

27889 

4657463 

12-9228480 

5-506879 

168 

28224 

4741632 

12-9614814 

5-517848 

169 

28561 

4826809 

13-0000000 

5-528776 

170 

28900 

4913000 

13-0384048 

5-539668 

171 

29241 

5000211 

13-0766968 

5-650499 

ITS 

29584 

5008448 

13-1148770 

5*661298 

173 

29929 

5177717 

13-1529464 

5-672054 

174 

30276 

5268024 

13-1909060 

5-582770 

175 

30626 

5359375 

13-2287666 

5-593445 

176 

30976 

5451776 

13-2664992 

5-604079 

177 

31329 

5545233 

13-3041347 

5-614673 

178 

31684 

5639752 

13-3416641 

5-625226 

179 

32041 

5736339 

13-3790882 

5-636741 

ISO 

32400 

6832000 

13-4164079 

5-646216 

181 

32761 

6929741 

13-4636240 

6-666662 

182 

33124 

6028568 

13-4007376 

6-667061 

183 

33489 

6128487 

13-6277493 

6-677411 

184 

33866 

6229604 

13*6646600 

6-687734 

186 

34226 

6331626 

13-6014706 

6-698019 

186 

34696 

6434866 

13-6381817 

6-708267 

187 

34969 

6639203 

13-6747943 

5-718479 

188 

36344 

6644672 

13-7113092 

6-728664 

189 

36r21 

6761269 

13-7477271 

6-738794 

190 

36100 

6869000 

13-7840488 

6-748897 

191 

36481 

6967871 

13-8202760 

6-768966 

192 

36864 

7077888 

13-8664066 

6-768998 

193 

37249 

7189067 

13-8924440 

6-778996 

194 

37636 

7301384 

13-9283883 

6-788960 

196 

38026 

7414876 

13-9642400 

5-798890 

196 

38416 

7629636 

14-0000000 

6-808786 

197 

38809 

7646373 

14-0366688 

6-818648 

198 

39204 

7762392 

14-0712473 

6-828476 

199 

39601 

7880699 

141067360 

6-838272 

200 

40000 

8000000 

14-1421366 

5-848036 

201 

40401 

8120601 

141774469 

6-867766 

202 

40804 

8242408 

14-2126704 

6-867464 

203 

41209 

8366427 

14-2478068 

6-877130 

204 

41616 

8489664 

14-2828569 

5-886766 

p 

'377 

^ 

^ 

Number. 

Square. 

Cube. 

Square  Boot 

Cube  Root. 

■ 

206 

42025 

8615126 

14-3178211 

6-896368 

206 

42430 

8741816 

14-3627001 

5-906941 

^^^H 

207 

42849 

8869743 

14-3874940 

5-915481 

^^^H 

208 

43264 

8908012 

14-4222051 

5-024902 

^^^H 

200 

43681 

9129329 

14  4568323 

5-934478 

^^^^1 

210 

44100 

9-261000 

14-4013767 

5-943922 

^^H 

211 

44521 

9393931 

14-5258390 

6-953341 

^^H 

212 

44944 

0528128 

14-5602108 

5-062731 

^^^^1 

213 

45369 

9063597 

14  ■5045195 

.'>-9720e2 

^^^H 

214 

45790 

0800344 

14-6-287308 

6-081426 

215 

46225 

9038375 

14-6628783 

6-090727 

^^^H 

2ia 

46656 

J 0077696 

14-6960385 

6-000000 

^^^H 

217 

47089 

102I8313 

14-7300109 

6000244 

V 

218 

47524 

10360232 

14-7648231 

6-018463 

219 

47961 

10503459 

14-7986486 

6027660 

1 

220 

40400 

10648000 

14-8323070 

0-036811 

^^^J 

221 

48841 

10793861 

14-8660687 

a-045943 

^^^^1 

222 

49284 

10041048 

14-8996644 

6055048 

^^H 

223 

49729 

110B9567 

14'9331845 

6-064120 

50176 

11230424 

14-9066295 

6-073177 

^^1 

^B£- 

50625 

11300625 

15-0000000 

6  082201 

^^B 

51076 

11543176 

150332064 

6081100 

^^^^1 

■Ip 

51529 

11607083 

16-(I665102 

6-100170 

^^^^1 

mf 

ai9B4 

11852352 

150096680 

6100115 

^^^^1 

22S 

62441 

12008989 

151327460 

6-118032 

^^^^1 

230 

52900 

12167000 

16- 1657609 

0-126025 

^^^^1 

231 

68361 

123263JI1 

15- 1986842 

6-135702 

^^^^M 

232 

53824 

12487168 

15-2315462 

6-144634 

^^^^M 

233 

54288 

12640337 

152643375 

6-153448 

^^^^H 

234 

64756 

12812904 

16-2970686 

6-162-239 

^^^^H 

236 

56226 

12977875 

16-3297007 

6- 17 1005 

^^^^H 

238 

65696 

13144256 

16-3622015 

6-170747 

237 

66160 

13312053 

15-3048043 

6-1B8463 

^^^^1 

238 

66644 

13481272 

15-4272486 

6-107164 

^^^^1 

238 

67121 

13651010 

16-4596248 

6-205H2i 

^^^^H 

240 

67600 

13824000 

16-4910331 

6214464 

^^^^1 

241 

58081 

] 3997621 

16-5241747 

6-223084 

^^^^1 

242 

58564 

14172488 

16-5563492 

6231 678 

^^^^M 

243 

59049 

14348007 

I5'5884673 

6240261 

^^^^M 

244 

59536 

14526784 

15-6204094 

0248800 

245 

60026 

14706125 

15-0524768 

6257a24 

^^^^M 

246 

60516 

14»8«036 

150843871 

6265826 

^^^^M 

247 

61000 

16060223 

16-7162336 

6-274304 

^^^^M 

248 

61504 

15252092 

15-7480167 

6282760 

^^^H 

24U 

62001 

16438240 

15-7797338 

6-291194 

^^^H 

260 

62600 

15625000 

15-8113883 

6-200004 

^^^H 

251 

«3001 

15813261 

15-8420705 

0-3y7O02 

^^^H 

252 

63504 

16003008 

15-8745079 

6316359 

^^^H 

2sa 

»1009 

16194277 

16-9060737 

6-324704 

^^^H 

264 

6461S 

16387064 

16-0373775 

6-333()25 

^^^H 

26S 

65025 

16581375 

15-9687194 

C-341325 

1 

J 

378 


Nimbar. 

Square. 

Cube. 

SqpiareBoor. 

Cube  Root 

256 

65636 

16777216 

^■[;ci  i  i  1  1  K^^ 

6*349002 

267 

66049 

16974593 

16-0312196 

6-357858 

268 

66564 

17173612 

16-0623784 

6-366086 

258 

67081 

17373979 

16H)934769 

6-374310 

200 

67600 

17676000 

16-1246166 

6-382604 

2ei 

68121 

17779681 

16-1664944 

6-300676 

262 

68644 

17984728 

16-1864141 

6-388827 

263 

69169 

18191447 

16-2172747 

6-406858 

264 

69696 

18399744 

16-2480768 

6-416068 

265 

70225 

18609625 

16-2788206 

6-423167 

266 

70766 

18821096 

16-3095064 

6-431226 

267 

71289 

19034163 

16-3401346 

6-438276 

268 

71824 

19248832 

16*3707066 

6-447306 

269 

72361 

19465109 

16*4012196 

6-456314 

270 

72900 

19683000 

16*4316767 

6*463304 

271 

73441 

19902611 

16-4620776 

6-471274 

272 

73964 

20123648 

16*4924226 

6-478224 

273 

74529 

20846417 

16*62*27116 

6-487163 

274 

75076 

20670824 

16*6529464 

6-486064 

276 

75625 

20796876 

16*5831240 

6-602956 

276 

76176 

21024576 

16-6132477 

6*610828 

277 

76729 

21263933 

16*6433170 

6-618684 

278 

77284 

21484952 

16*6733320 

6-626518 

279 

77841 

21717639 

16*7032931 

6-634336 

280 

78400 

21962000 

16*7332006 

6-642132 

281 

78961 

22188041 

16*7630546 

6-548811 

282 

79524 

22425768 

16*7928556 

6-557672 

283 

80089 

22665187 

16-8226038 

6-565415 

284 

80656 

22906304 

16-8522995 

6-573139 

285 

81225 

23149125 

16*8819430 

6-580844 

286 

81796 

23393656 

16*9115345 

6-588531 

287 

82369 

23639903 

16-9410743 

6-596202 

288 

82944 

23887872 

16-9705627 

6*603854 

289 

83521 

24137569 

17-0000000 

6*611488 

290 

84100 

24389000 

17-0293864 

6*619106 

291 

84681 

24642171 

17-0587221 

6-626705 

292 

85264 

24897088 

170880075 

6-634287 

293 

85849 

25153757 

171172428 

6-641851 

294 

86436 

25412184 

171464282 

6-649399 

295 

87025 

25672375 

171755640 

6-656930 

296 

87616 

25934336 

17-2046505 

6*664443 

297 

88209 

26198073 

17-2336879 

6671940 

298 

88804 

26463592 

17-2626765 

6-679419 

299 

89401 

26730899 

17-2916165 

6-686882 

300 

90000 

27000000 

17-3205081 

6-6943*28 

301 

90601 

27270901 

17-3493516 

6-701759 

302 

91204 

27543608 

17-3781472 

6-709172 

303 

91809 

27818127 

17-4068952 

6-716569 

304 

92416 

28094464 

17-4355958 

0-723950 

305 

93025 

28372625 

17-4642492 

6-731316 

306 

93636 

28652616 

17-4928557 

6-738665 

p 

■ 

379 

^ 

^ 

Number. 

Sqoaie. 

Cube- 

Squnre  Root. 

Cube  Root 

■ 

307 

04249 

28934443 

17-5214155 

0-745097 

303 

948e4 

29218 112 

17-5400288 

0-733313 

^^^H 

800 

05481 

29503629 

17-5783958 

6-760014 

^^^H 

310 

08100 

29791000 

17-6068169 

6-767809 

^^^H 

311 

06721 

30080231 

17-6351921 

6-77316B 

312 

97344 

30371328 

17-6(135217 

6-782422 

^^^1 

313 

07909 

30664297 

17-6018000 

6-789661 

^^H 

314 

0(1596 

30959144 

17-7200451 

6-796884 

31S 

99225 

31265875 

17-7482393 

6-804091 

^^^H 

818 

09856 

31564406 

17-7763888 

6-811284 

^^^H 

317 

100489 

31855013 

17-8044038 

6-818461 

^^^^1 

31U 

10U24 

32157432 

17-8325545 

6-825624 

^^^^M 

31» 

101761 

32461750 

17-8003711 

6-832771 

^^^^M 

320 

102400 

32768000 

17-8885438 

6-8:W903 

^^^^1 

321 

103041 

33076101 

17-9164729 

6-847021 

^^^^1 

322 

1036B4 

83380248 

17-9443584 

6-854124 

323 

104329 

33008267 

17-9722008 

6-861 211 

^^^^H 

324 

104976 

34012224 

18-0000000 

0-868286 

^^^^H 

32d 

103626 

34328125 

18-0277564 

6-876343 

^^^^H 

820 

106276 

34045976 

18-0554701 

6-882388 

^^^^H 

327 

100029 

34005783 

18-0831413 

6-889419 

^^^^1 

326 

107584 

36287552 

18-1107703 

6-800435 

^^^^1 

32» 

108241 

36611289 

18-1383571 

6-903436 

330 

108900 

35037000 

)8-I659021 

6-910423 

^^^H 

331 

109561 

36264091 

181934054 

6-917396 

332 

110224 

365943(i8 

18-2208672 

6-924355 

^^^^1 

333 

110889 

36026037 

18-2482870 

6-931300 

^^^^1 

334 

111550 

37250704 

18-2756669 

0036232 

^^^^1 

33& 

112225 

37596375 

18-3030052 

0045149 

^^^^1 

336 

112896 

37933036 

18-3303028 

6-952063 

^^^^1 

887 

113569 

38272753 

18-3673598 

6-958943 

^^^^1 

336 

114244 

38014472 

18-3847703 

6-965819 

^^^H 

33B 

114921 

3H968219 

18-4 11 96-20 

6-972682 

340 

115600 

39304000 

18-4390889 

6-079532 

^^^H 

341 

116281 

30661821 

18-4661853 

6-986300 

^^^^1 

342 

116964 

40001088 

18-4032420 

6-993191 

^^^^H 

843 

117649 

40333607 

18-6202592 

7-000000 

^^^^H 

344 

118336 

40707584 

1 8-547-2370 

7-006706 

^^^^H 

346 

119025 

41063625 

18-6741756 

7-013570 

^^^^H 

346 

110716 

41421736 

18-6010752 

7-020349 

^^^^1 

at7 

120409 

41781923 

18-0270360 

7-027106 

^^^^1 

348 

12 1104 

42144102 

18-6547581 

7-033850 

^^^^1 

349 

121801 

42508349 

18-8815417 

7-040581 

3-^0 

122600 

42875000 

18-7082860 

7-047208 

^^^H 

351 

123201 

43243561 

18-7340040 

7-054003 

352 

123904 

43614208 

18-7616630 

7-060606 

^^^^H 

353 

124609 

43986077 

18-788-2942 

7-067376 

^^^^H 

354 

125316 

44361804 

18-8148877 

7-074O43 

^^^^H 

355 

126025 

44738875 

18-8414437 

7-080698 

^^^^H 

35« 

126730 

46118010 

18-8670023 

7-087341 

^^^^H 

367 

127449 

46490203 

7-093070 

1 

M 

380 


Nuniber. 

Square. 

Cube. 

Square  Root 

Cube  Root. 

a58 

128164 

45882712 

18-9208870 

7-100588 

359 

128881 

46268279 

18-0472063 

7-107198 

360 

120600 

46656000 

18-0736660 

7-118786 

361 

130321 

47045881 

10-0000000 

7*120867 

362 

131044 

47437928 

10-0262076 

7'128885 

363 

131769 

47832147 

10-0626680 

7-138402 

364 

132496 

48228544 

10-0787840 

7-140087 

365 

133225 

48627125 

10-1040732 

7-146660 

366 

133956 

49027896 

10-1311266 

7*168000 

367 

134689 

49430863 

10-1672441 

7-168609 

368 

135424 

49836032 

10-1833261 

7-166095 

369 

136161 

50243409 

10-2003727 

7-172680 

370 

136900 

60653000 

10*2353841 

7-179064 

371 

137641 

51064811 

10-2613603 

7-186516 

372 

138384 

51478848 

10-2873016 

7-191966 

373 

139129 

51895117 

10-3132070 

7-108405 

374 

139876 

52313624 

10-3390796 

7-204882 

375 

140625 

52734375 

10-3640167 

7-211247 

376 

141376 

53157376 

10-3907194 

7-217652 

377 

142129 

63582633 

10*4164878 

7-224045 

378 

142884 

64010152 

10-4422221 

7-280427 

379 

143641 

64439939 

10-4679223 

7-236797 

380 

144400 

64872000 

19-4935887 

7-248156 

381 

145161 

65306341 

19-6102213 

7-249504 

382 

145924 

66742968 

10-6448203 

7-266841 

383 

146689 

66181887 

10-6703868 

7-262167 

384 

147456 

56623104 

10-5959179 

7-268482 

385 

148225 

57066625 

19-6214169 

7*274786 

386 

148996 

57512^156 

19-6468827 

7-281079 

387 

149769 

57960603 

19-67-23156 

7-287362 

388 

150544 

58411072 

19-6977156 

7-293688 

389 

151321 

58863869 

19-7230829 

7-209893 

390 

152100 

59319000 

19-7484177 

7-306148 

391 

152881 

59776471 

19-7737199 

7-312388 

392 

153664 

60236288 

19-7989899 

7-31B611 

393 

154449 

60698457 

19*824-2276 

7-324829 

394 

155236 

61162984 

19-8494332 

7-331087 

395 

156025 

61629875 

19-8746069 

7-387284 

396 

156816 

62099136 

19-8997487 

7-848420 

397 

157009 

62570773 

19-9248688 

7-349606 

398 

158404 

6:)044792 

19-9499373 

7-365702 

399 

159201 

63521199 

19-9749844 

7-361017 

400 

160000 

64000000 

20-0000000 

7-868068 

401 

160801 

64481201 

20*0249844 

7-374188 

402 

161604 

64964808 

20-0499377 

7-380822 

403 

162409 

65450827 

20-0748690 

7-886487 

404 

163216 

65939264 

20*0997612 

7:802542 

405 

164025 

66430125 

20-1246118 

7-888686 

406 

164836 

06923416 

20-1494417 

7-404720 

407 

165649 

67419143 

20*1742410 

7-410794 

408 

166464 

67917312 

20-1000009 

7*416869 

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timber. 

Squ«n>. 

Cube. 

Square  RooU 

Cube  Root 

k 

408 

167281 

68417!t29 

20-2237484 

7-422914 

■   410 

laaioo 

68921000 

20-2484567 

7-428958 

■  411 

108921 

6942a33l 

30-2731349 

7-434993 

■  412 

169744 

6993462B 

20-2977831 

7-441018 

■  413 

17066!) 

70444997 

20-3224014 

7-447034 

■  414 

171306 

70037944 

20-3460899 

T-453039 

■  416 

172225 

71473375 

20-3716488 

7-460036 

n 

416 

173060 

71991296 

20-3900781 

7-466022 

417 

173889 

72511713 

20-4205779 

7-47099J* 

41B 

174724 

73034032 

20-4450483 

7-476060 

419 

175661 

73560059 

20-4694895 

7-482924 

420 

176400 

74088000 

20-4939015 

7-488872 

441 

177241 

74618461 

20-5182846 

7-494810 

422 

178084 

75161448 

20-5126386 

7-500740 

423 

178020 

75686967 

20-5660638 

7-506660 

424 

179776 

76225024 

20-5912603 

7-512671 

^^^^H 

425 

180625 

76765625 

20-6156281 

7-518473 

^^^^H 

420 

181476 

77308776 

20-6397674 

7-524365 

^^^^H 

427 

102329 

77854403 

20-6639783 

7-530248 

^^ 

428 

183184 

78402752 

20-6881609 

7-636122 

429 

184041 

78963609 

20-7123162 

7-541086 

430 

184000 

79507000 

20-7364414 

7-547M2 

481 

185761 

80002901 

20-7605396 

7-653680 

432 

188024 

80021668 

20-7846097 

7-660526 

433 

187480 

81182737 

20-8086520 

4M 

188356 

81746504 

20-0326067 

7-571173 

436 

189226 

82312876 

20-8566636 

7-576984 

436 

190080 

82881866 

20-8806130 

7-582786 

437 

190969 

83453463 

20-9046460 

7-588570 

438 

191844 

84027672 

20-9284495 

7-594363 

430 

192721 

84604.J19 

20-0523208 

7-600130 

440 

103600 

06184000 

20-9761770 

7  ■606006 

441 

194481 

86706121 

21-0000000 

7  611662 

442 

1963C4 

86350888 

21-0237960 

7-617411 

443 

196249 

06930307 

21-0475052 

7-623161 

444 

197136 

87628384 

21-0713075 

7-620883 

445 

198025 

80121125 

21-0960231 

7-634606 

446 

lOtlOlO 

88716536 

211187121 

7-640321 

447 

109009 

80314623 

21-1423745 

7-646027 

448 

200704 

89915392 

21-1660106 

7-651725 

449 

201601 

90510849 

21-1806201 

7-057414 

460 

202500 

91126000 

21-2132034 

7-663094 

451 

203401 

91733051 

21  ■2367606 

7  6607tMi 

452 

204304 

92346408 

21-2002910 

7■67^430 

453 

205200 

92959677 

21-2837967 

7-6>tOOB6 

454 

206116 

93576664 

21-3072758 

7-«!6732 

465 

207025 

94196375 

21-3307200 

7091371 

450 

207936 

94818816 

21-3541565 

7  097002 

467 

208849 

95443993 

21-3775683 

7-702624 

M 

458 

209704 

96071912 

21 -4009340 

7-708230 

1 

45fl 

210G81 

06702579 

21-4242853 

7-713844 

1 

i 

^ 

i 

C  C 

m 

38@ 


Number. 

Square. 

Cube. 

Square  Root 

Cube  Root 

460 

211600 

97336000 

21-4476106 

7-719442 

461 

212521 

97972181 

21*4709106 

7-725082 

4G2 

213444 

98611128 

21*4941853 

7-730614 

463 

214369 

99252847 

21-5174348 

7-736187 

464 

215296 

99897344 

21-5406592 

7-741758 

465 

216225 

100544625 

21-5638587 

7-747310 

466 

217156 

101194696 

21-5870331 

7-752800 

467 

218089 

101847563 

21-6101828 

7-758402 

468 

219024 

102503232 

21-6333077 

7-763886 

469 

219961 

103161709 

21-6564078 

7-769462 

470 

220900 

103823000 

21-6794834 

7-774980 

471 

221841 

104487111 

21-7025344 

7-780480 

472 

222784 

105154048 

21-7255610 

7-785982 

473 

223729 

105823817 

21-7485632 

7-791487 

474 

224676 

106496424 

21-7715411 

7-796974 

476 

225625 

107171875 

21-7944947 

7-802458 

476 

226576 

107850176 

21-8174242 

7-807925 

477 

227529 

10B531333 

21-8403297 

7-813388 

478 

228484 

109215352 

21-8632111 

7-818846 

479 

229441 

109902239 

21-8860686 

7-824284 

480 

230400 

110592000 

21-9089023 

7-828786 

481 

231361 

111284641 

21-9317122 

7-835168 

482 

232324 

111960168 

21-9544984 

7-840684 

483 

233289 

112678587 

21-9772610 

7-84601^ 

484 

234256 

113379904 

22.0000000 

7-851424 

485 

235225 

114084125 

220227155 

7-856828 

486 

236196 

114791256 

22-0454077 

7-8622-24 

487 

237 169 

115501303 

22-06807a5 

7-867613 

488 

238144 

116214272 

22-0907220 

7-872984 

489 

239121 

1 16930169 

22- 1133444 

7-878368 

490 

240100 

117649000 

22- 1359436 

7-883735 

491 

241081 

118370771 

221585198 

7-889095 

492 

242064 

1 19095488 

22- 1810730 

7-894446 

493 

243049 

119823157 

22-2036033 

7-899791 

494 

244036 

120553784 

22-2261108 

7-905129 

495 

245025 

121287375 

22-2485955 

7-910460 

496 

240016 

122023936 

22-2710575 

7-915784 

497 

247009 

122763473 

22-2934968 

7-921100 

498 

248004 

123505992 

22-3159136 

7-926408 

499 

249001 

124251499 

22-3383079 

7-931710 

500 

250000 

125000000 

22-3606798 

7-937005 

501 

251001 

125751501 

22-3830293 

7-942293 

502 

252004 

126506008 

22-4053565 

7-947573 

503 

253009 

127263527 

22-4276615 

7-952847 

504 

254016 

128024064 

22-4499443 

7-958114 

505 

255025 

128787625 

22-4722051 

7-963374 

506 

256036 

129554216 

22-4944438 

7-968627 

507 

257049 

130323843 

22-5166605 

7-973873 

508 

258064 

131096512 

22-5388553 

7-979112 

509 

259081 

131872229 

22-5610283 

7-864344 

510 

260100 

132651000 

22-5831796 

7-868668 

V                383                     ^^H 

Ifembe,. 

SqU«e, 

Cube. 

Square  Root 

Cube  Root. 

511 

2B1121 

133432831 

22-8053091 

7-994788 

612 

262144 

134217728 

22-6274170 

8-000000 

&13 

■263100 

135005697 

22-64y5033 

8005205 

614 

264190 

135796744 

22-6715681 

8-0HM03 

615 

26J2-25 

136590875 

22-6036114 

8-016595 

6ie 

266266 

137388090 

22-7156334 

8-020779 

517 

267289 

138188413 

22-7376340 

8-025057 

518 

268324 

138991832 

22-7690134 

8-031120 

510 

260361 

139798350 

22-7816715 

8-036293 

520 

270400 

140608000 

22-8035085 

8-04 1451 

521 

271441 

141420701 

22-8254244 

8040603 

523 

272484 

142236648 

22-8473103 

8051748 

523 

273520 

143056607 

22-8601033 

8-056886 

^^^^M 

524 

274576 

143877824 

22-8010463 

8-082018 

^H 

525 

275625 

144703126 

22-9128785 

8-067143 

^H 

526 

276076 

145531576 

22-9346899 

8-072262 

527 

277729 

146363183 

22-0664806 

8-077374 

H 

628 

278784 

147197952 

22-9782606 

8-082480 

1 

628 

270841 

148035889 

23-0000000 

8-087579 

630 

280000 

148877000 

23-0217289 

8-002672 

I 

631 

281961 

140721201 

230434372 

8-007758 

1 

532 

283024 

160568768 

23-06512&2 

8-1028:K» 

1 

533 

284089 

151419437 

23-0867928 

8- 107912 

J 

534 

285156 

152273304 

23-1084400 

8-112980 

^^^H 

636 

286225 

153130375 

231300670 

8118041 

^^^^1 

636 

287296 

153900656 

231516738 

8123096 

^^^^1 

537 

288360 

154854153 

231732005 

8-128144 

^^^^1 

538 

289444 

165720872 

231948270 

8133186 

^^^^1 

539 

290521 

150590819 

23-2163736 

8-138223 

^^^^1 

540 

201600 

157464000 

23-2379001 

8*143253 

^^^^1 

641 

202G81 

158340421 

23-2504067 

8-140276 

^^^^1 

642 

293764 

159220088 

23-2808935 

8-153293 

643 

294849 

160103007 

23-3023604 

8-158305 

644 

295936 

160980184 

23-3238076 

8-163309 

646 

287025 

161878625 

23  345-J351 

8-168309 

646 

288110 

162771336 

23-3666429 

8.173302 

547 

209209 

163667323 

23-3880311 

8-178209 

548 

300304 

1 04566592 

23-4093988 

8-183260 

640 

301401 

105469 J 49 

23-4307490 

8-188244 

560 

302500 

166375000 

23-4520788 

8-193212 

661 

303001 

167284151 

23-4733892 

8-198175 

662 

304704 

lfi81!MMJ08 

23-4946802 

8-203131 

553 

305800 

169H2377 

23-5150520 

8-208082 

654 

306016 

170031404 

23-5372046 

8-213027 

555 

308025 

170953875 

23-6584380 

8-217965 

550 

3001S6 

171870616 

23-5796522 

8-22-2898 

557 

310249 

172808693 

23-6008474 

H-227825 

»s 

311304 

173741112 

23-6220236 

8-232746 

65S 

312481 

174676879 

23-6431808 

8-237661 

^  MO 

313600 

175616000 

23-6643191 

8-242670 

If 

.»i 

314721 

176668481 

23-6864386 

8-247474 

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1 

m 

CC  2 

M 

384 


1 

Number. 

Square. 

Cube. 

Square  Root. 

CubeBooL 

502 

315844 

177504328 

23-7065392 

8-262371 

563 

316969 

178453547 

23-7276210 

8-267268 

564 

318096 

179406144 

23-7486842 

8*282140 

5(J5 

319225 

180362125 

23*7607286 

8-267029 

566 

320356 

181321496 

23-7907545 

8-271903 

567 

321489 

182284263 

23-8117618 

8-276772 

56(i 

322624 

183250432 

23-8327506 

8-281636 

569 

323761 

184220009 

23-8537209 

8-286493 

570 

324900 

185193000 

23-8746728 

8-291344 

571 

32(M)41 

186169411 

23-8956063 

8-296190 

572 

327  UM 

187149248 

23-9165215 

8-301030 

573 

328329 

188132517 

23-9374184 

8-306866 

574 

329476 

189119224 

23-9582971 

8-310094 

575 

330625 

190109375 

23-9791576 

8-316617 

576 

331776 

191102976 

240000000 

8-320336 

577 

3:)2929 

192100033 

24-0208243 

8-326147 

570 

334084 

193100552 

24*0416306 

8-329964 

579 

33524 1 

194104539 

2406-24188 

8-334766 

5H0 

336400 

195112000 

24  0831892 

8-339661 

581 

337561 

196122941 

24-1039416 

8-344341 

582 

338724 

197137368 

24- 1246762 

8-349126 

583 

33JW89 

198155287 

24- 1453929 

8-363904 

M4 

341056 

199176704 

24-1660919 

8-368678 

585 

342225 

200201625 

24-1867732 

8-363446 

586 

343396 

201230056 

24-2074369 

8-368209 

587 

^44569 

202262003 

24-2280829 

8-372960 

588 

345714 

203297472 

24-24^)7113 

8-377718 

589 

34(K>21 

2043:Mi4(>9 

24-2(>932*22 

8-382466 

59<) 

348100 

205379000 

24-281>9156 

8-3B7206 

591 

.  349281 

206425071 

24-3104916 

8-391942 

692 

350464 

2074746i{8 

24-3310501 

8-306673 

593 

351649 

208527857 

24-3^315913 

8-401308 

594 

35283(J 

209584;jii4 

24-3721152 

8-406118 

595 

354025 

210644875 

2^4-3926218 

8-410832 

59<; 

355216 

211708736 

24-4131112 

8-416642 

597 

356409 

'  212776173 

24-4335834 

8-420246 

598 

357604 

213847192 

24-4540385 

8-424044 

599 

358801 

21492175)9 

24-4744765 

8-420638 

600 

3(;0000 

216000000 

24-4948974 

8-434327 

601 

36] 201 

217081801 

24-515;M)13 

8-430000 

602 

{¥12404 

218167208 

24-5356883 

8-443687 

603 

IVGatiOU 

219256227 

24-5560583 

8-448360 

604 

364816 

220348864 

24-5764115 

8*463028 

605 

366025 

221445125 

24-5967478 

8-467600 

(506 

MTlim 

222545016 

24-6170673 

8-162347 

607 

368449 

223648543 

24-6373700 

8-466000 

608 

lUfmG4 

224755712 

24-6576560 

8-471647 

609 

370881 

2258(;f{529 

24-6779264 

8-476289 

610 

372100 

226981000 

24-6081781 

8-480996 

611 

373321 

22^K)99131 

24-7184142 

fr486667 

612 

374544 

229220928 

24-7386338 

8-400184 

SS5                                                        ^^H 

liimbci. 

gqu«^ 

Cube. 

Square  RooL 

Cube  Root 

■ 

613 

375769 

230346307 

24-7588308 

8-494806 

014 

376006 

231475544 

24-7790234 

8-400423 

^^^^^1 

015 

378-225 

23260(1375 

24-7991036 

8-5O4035 

^^^^^1 

010 

370456 

233744806 

24-8193473 

8-508641 

^^^^^1 

CI7 

3«061t(l 

234805113 

24-8394847 

8-513243 

^^^^H 

618 

301J<24 

236029032 

24-8596058 

8-517840 

^H 

619 

383101 

23717B050 

24-8797100 

8-522432 

^1 

020 

384400 

23032800<l 

24-8997992 

8-.527018 

^^^^H 

631 

365041 

23II4H3061 

24-9108716 

8531600 

022 

38011(14 

240641848 

24-9390278 

8-630178 

^^^^H 

m-A 

388 ISO 

241804367 

24-9699679 

8-640750 

^^^^H 

6-24 

380370 

242070624 

24-9799920 

8-545317 

^^^^H 

025 

31KI625 

244140625 

260O000OO 

8-649870 

^^^^H 

020 

391876 

245314376 

230199920 

8-554437 

^^^^H 

627 

393 12U 

240401883 

250309681 

8-658900 

^^^^H 

828 

304384 

247673162 

25-0599282 

8-563537 

^^^^H 

020 

30.5641 

241)858189 

26-0798724 

8-668080 

^^^^H 

630 

30C»I>0 

250047000 

260998008 

8-672618 

^^^^H 

031 

308101 

251239591 

25-1107134 

8-577152 

032 

399424 

25243-5968 

26-1396102 

8-581680 

^^^^H 

033 

400080 

253636137 

261594013 

8-586204 

^^^^H 

034 

401U50 

254840104 

261793566 

8-590723 

^^^^H 

035 

403225 

250047875 

25U)92063 

8-596238 

^^^^H 

030 

404496 

257259456 

25-2190404 

8-509747 

^^^^H 

037 

405769 

258474853 

25-2388580 

8-604252 

^^^^H 

038 

407044 

259604072 

2&-2586ei9 

8008752 

^^^^H 

03(t 

408321 

260917119 

25-2784493 

U-0 13246 

^^^^H 

640 

409000 

202144000 

25-298-2213 

8-617738 

^M 

041 

410881 

203374721 

25-3179778 

8-622224 

H 

642 

412164 

264609288 

25-3377180 

8-0-26706 

643 

413440 

265847707 

2.5-3574447 

8-631183 

^^^^H 

644 

414736 

267089084 

25-377155I 

8-035055 

^^^^H 

645 

416025 

208336125 

25-3908502 

8-640122 

^^^^H 

640 

417316 

269580136 

25-4165301 

8044585 

^^^^H 

647 

418009 

270840023 

26-4361947 

804H043 

^^^^H 

648 

419904 

272097792 

26-4558441 

H053497 

^^^^H 

049 

421201 

273359440 

26-4754784 

8-067946 

^^^^H 

650 

422500 

274025000 

25-4950970 

8-662301 

^^^^H 

((51 

423801 

275894451 

25-5147010 

8-666831 

652 

425104 

277167808 

25-6342907 

8671266 

^^^^H 

653 

426409 

278445077 

25-5538647 

8675697 

^^^^H 

054 

427716 

279726264 

255734237 

8-680123 

^^^^H 

655 

420025 

28101 1375 

25-5029078 

8-684545 

^^^^H 

050 

430336 

282300416 

25-0124069 

8-688963 

^^^^H 

067 

431649 

203503393 

25-0320112 

R-693376 

^^^^H 

058 

432St04 

281890312 

25-0515 107 

8-697784 

^^^^H 

«6!l 

434281 

280191170 

25-6709933 

0-702 188 

^^^^H 

600 

435600 

28J49fH)00 

25-(t9O4052 

8706587 

001 

436921 

S8B8047B1 

25-700i)203 

8-710982 

^^^^H 

^_ 

1  062 

43i(244 

290117628 

25-7293607 

8  715373 

^^^^H 

1 

F. 

439569 

291434247 

26-7487864 

8-710750 

^^H 

1 

1 

^ 

k      m 

■ 

38G 


Number. 

Square. 

Cube. 

Cube  Root 

G04 

440890 

292754044 

26-7681975 

8-724141 

065 

442225 

294079025 

25-7875939 

8-728518 

060 

443550 

295408290 

26-8069768 

8-732891 

007 

444889 

290740903 

26-8263431 

8-737200 

008 

440224 

298077032 

26-8466960 

8-741624 

009 

447501 

2fl9418309 

26-8660343 

8-746984 

070 

448900 

30O7e:JO00 

26-8843582 

8-750340 

071 

450241 

302111711 

26-9030077 

8-764691 

672 

451584 

303404448 

26-9229028 

8-769038 

073 

452929 

304821217 

26-9422436 

8-763380 

074 

454276 

300182024 

26-9616100 

8-767719 

076 

455625 

307540875 

25-9807021 

8-772058 

070 

450976 

308915770 

260000000 

8-776382 

077 

458329 

310288733 

26-0192237 

8-780708 

078 

459684 

311005752 

26-0384331 

8-785029 

079 

461041 

313040839 

26-0670284 

8-789346 

080 

462400 

314432000 

200768096 

8-793659 

081 

403701 

315821241 

260969707 

8-797967 

082 

405124 

3ir214568 

261151297 

8-802278 

083 

400489 

318611987 

26-1342687 

8-806672 

084 

407850 

320013504 

26*1533937 

8-810868 

085 

409225 

321419125 

26-1726047 

8-815169 

080 

470590 

322828850 

261916017 

8-819447 

087 

471909 

324242703 

26-2106848 

8-823790 

(m8 

473344 

325000072 

26-2297541 

8-828009 

089 

474721 

327082709 

26-2488()95 

8-832286 

WK) 

470100 

328509000 

26-2678^511 

8-836666 

091 

477481 

329939371 

26-2868789 

8-840822 

092 

47«8f>4 

3313738(m 

26-305lW)29 

8-846085 

093 

4H0249 

332812557 

20-3248932 

8-849344 

094 

481030 

3342553^4 

26-343in97 

8-853698 

095 

483025 

333702375 

26-3628527 

8-867849 

690 

484410 

337153530 

26-3818119 

8-862096 

097 

485809 

338008873 

26-4007570 

8-866337 

098 

487204 

340008392 

20-4196890 

8-870676 

099 

488001 

341532099 

20-4:^(0081 

8-874809 

700 

4JK)(K)0 

343000000 

20-4575131 

8-879040 

701 

491401 

344472101 

20-4704046 

8-883266 

702 

492804 

345948408 

20-4952826 

8-887488 

703 

494209 

347428927 

26-5141472 

8-891706 

704 

495010 

:M89  13004 

26-5329983 

8-896920 

705 

497025 

350402025 

26-5618361 

8-900130 

700 

498430 

351895810 

26-5700606 

8-904336 

707 

499849 

353393243 

20-5894716 

8-908638 

708 

501204 

3548SM912 

26-6082094 

8-912736 

709 

502081 

350400829 

26-6270639 

8-916981 

710 

504100 

357911000 

26-6468262 

8-921121 

711 

505521 

359425431 

26-6646833 

8<9259(f} 

712 

500944 

360944128 

26-6833281 

8-989490 

713 

508360 

362467007 

26-7020606 

8-98800B 

714 

509796 

363894344 

26-7207784 

8«m48 

1 

p 

387 

^ 

Kunbu. 

Squue. 

Cube. 

Square  RooL 

Cube  Root. 

715 

611225 

365525875 

26-7394«39 

8-942014 

71« 

612656 

367061696 

26-7581763 

8-946180 

717 

614089 

3680U1813 

267768557 

8-950343 

718 

615524 

370146232 

26-7965220 

8-954502 

719 

616061 

371094969 

36-8141734 

8-958658 

7ai 

61t)4U0 

373248000 

268328157 

8-062809 

721 

51»«41 

374805361 

260514432 

8-900957 

722 

6212H4 

376367048 

26-8700577 

8-971100 

723 

622720 

377033067 

268886503 

8-975240 

724 

624170 

379603424 

26  9072481 

8-079370 

73& 

626025 

38107B125 

26W58240 

8-983508 

720 

62707fi 

382657176 

20-9443872 

8-087037 

727 

62862U 

384240583 

26-9629375 

8-001702 

728 

620984 

385828352 

269814751 

8-095883 

729 

631441 

387420489 

270000000 

9-000000 

730 

532800 

309017000 

270185122 

9-OIMl  13 

731 

534U<il 

390617891 

27-03701 17 

9-008-2-22 

732 

335824 

392223108 

270554985 

9-012328 

733 

637289 

303832837 

27-073!t727 

0-016430 

734 

538756 

305446904 

27-0924344 

0-020529 

736 

540225 

397065375 

27-1108834 

9024623 

730 

541006 

398688250 

27-1293199 

9-028714 

737 

543169 

400315553 

27-1477439 

0-032802 

73B 

544644 

401947272 

27-1601554 

9-036885 

739 

546I2I 

403583419 

27-1846544 

9-040966 

740 

647000 

406224000 

27-2029410 

0-045041 

741 

540081 

406860021 

27-2213152 

0-040114 

742 

650564 

408518488 

27-2300760 

9-053183 

743 

662049 

410172407 

27-2580263 

0-057248 

744 

563536 

411830784 

27-2703634 

9-061309 

746 

556025 

413493625 

27-2940881 

9'0G5367 

748 

556616 

415160036 

27-3130006 

9-060422 

747 

S68009 

416832723 

27-3313007 

9-073472 

748 

56S604 

418508902 

27-3495887 

9-077310 

749 

501001 

420180749 

27-3678644 

0-081563 

750 

562500 

421875000 

27-3861279 

9-0856113 

761 

564001 

423664751 

27-40437!t2 

9-089639 

762 

5B6504 

425250008 

27-4-226184 

9-003672 

763 

507009 

426957777 

27-4408455 

0-007701 

764 

6686 1« 

428061064 

27'459O604 

9-101726 

766 

570026 

4303611876 

27-4772633 

9-105748 

750 

571536 

432081216 

9-100766 

767 

573049 

433708093 

27-5136330 

9-113781 

758 

574504 

435510512 

27-5317908 

9-117793 

759 

576081 

437245479 

27-5400546 

0121801 

700 

577000 

438976000 

27-5080975 

9- 1-25805 

701 

6791-21 

440711001 

27-5802284 

0-120806 

702 

5806-14 

442450728 

27-0043476 

0-133803 

703 

6H21C1I 

444194047 

27-6224546 

0-137707 

1 

704 

583696 

445943744 

27-640341W 

0-1417»«! 

■^ 

6S5225 

447697125 

27-6586334 

9- 145774 

1 

L 

1 

B 

388 


Number. 

Square. 

Cube. 

Square  Root 

Cube  Root 

766 

686766 

449466096 

27-6767060 

9-149757 

767 

68^(289 

461217(563 

27-6947648 

9-153737 

76B 

681Mi24 

462984832 

27-7128129 

9-167718 

760 

691361 

464766609 

27-7308492 

9-161686 

770 

692900 

466633000 

27-7488739 

9-165656 

771 

694441 

468314011 

27*7668868 

9-169622 

772 

696984 

460099048 

27-7848880 

9*173586 

773 

697629 

46U(89917 

27-8028776 

9-177544 

774 

699076 

463684824 

27-8208666 

9*181600 

776 

600626 

466484376 

27-8388218 

9-185462 

776 

602176 

407288676 

27-8667766 

9-189401 

777 

603729 

46909743;^ 

27-8747197 

9-193347 

778 

606284 

470910962 

27-8926614 

9-197289 

779 

606841 

472729139 

27-9106716 

9-201228 

780 

608400 

474662000 

27-9284801 

9-205164 

781 

609961 

476379641 

27-9463772 

9*209096 

782 

611624 

478211768 

27-9642629 

9*213026 

783 

613089 

480048687 

27-9821372 

9-216960 

784 

614&i6 

481890304 

280000000 

9-220872 

786 

6l622i> 

483736626 

28-0178616 

9-224791 

786 

617796 

486687666 

28-0366916 

9-228706 

787 

619:)69 

487443403 

280636203 

9-232618 

788 

620944 

489303872 

28-0713377 

9-236627 

789 

622621 

491169069 

28*0891438 

9.240433 

790 

024 100 

49;K)3JK)00 

28-1069386 

9-244336 

791 

626681 

4iHin:\(il\ 

28-1247222 

9-248234 

792 

627264 

490793088 

28- 1424946 

9*252130 

793 

028849 

498677267 

28-1602667 

9-256022 

794 

630436 

6(K)666Ui4 

•i8-178006(J 

9-259911 

796 

032026 

602465)876 

•28-1967444 

9-263797 

796 

6330  k; 

6043o83:J6 

28-2134720 

9-267679 

797 

636209 

6002(U673 

28-23118^^4 

9-271569 

798 

630804 

608169692 

28-2488938 

9-276436 

799 

63)U01 

610082399 

28-20({6881 

9-279308 

800 

640000 

612000000 

28-2842712 

9-283177 

801 

641601 

613022401 

28-3019434 

9-287044 

802 

643204 

616849608 

28-3196046 

9-290907 

803 

644809 

517781027 

28-3372646 

9-294767 

804 

6404 {6 

6U)718404 

28-364iU>38 

9-296623 

806 

648026 

621600126 

28-3726219 

9-302477 

806 

G4iHim 

623(J06(n6 

28-3901391 

9-306327 

807 

061249 

626667943 

28-4077464 

9-310176 

808 

6628(U 

627614112 

28-4263408 

9-314019 

809 

664481 

629475129 

28-4429263 

9-317859 

810 

666100 

63144 1(K)0 

28-4604989 

9-321697 

8il 

667721 

63aill73l 

28-4780617 

9*325632 

812 

669344 

635387328 

28-4966137 

9-329363 

813 

660969 

637366797 

28-6131549 

9-338191 

814 

6(J2696 

639353144 

28-6306852 

9-337016 

816 

664226 

641343376 

28.5482048 

9-340838 

816 

666866 

643338496 

28-5653137 

9-344667 

1 

P 

■ 

380 

^ 

IT 

Niuober. 

Square. 

Cube. 

Square  Root 

Cube  Root 

817 

867409 

645338513 

28&832119 

9-348473 

8ia 

669124 

647343433 

28-6006903 

9-36-2285 

819 

6707C1 

549353250 

28-6  IB  1760 

9-356095 

II 

820 

C72400 

551308000 

28-6366421 

9-350901 

m. 

8<il 

674041 

5533B7061 

28-ftj30076 

9-363704 

■ 

822 

675684 

555412248 

28070542-1 

0-307505 

■ 

823 

67732B 

667441767 

280a797(i6 

9-371302 

■ 

824 

678076 

659476224 

28-7054002 

9-375096 

r 

825 

080623 

561516625 

28-7228132 

!»-378887 

840 

683276 

563650076 

28-7402157 

9-302<t75 

827 

083920 

50560!)283 

28-7570077 

9-386400 

828 

685584 

567663562 

28-7749891 

9-390241 

829 

687241 

569722780 

28-7923601 

9-394020 

^^^^1 

830 

688000 

571787000 

28-8097206 

9-397796 

^^^^1 

831 

600561 

573856191 

28-8270706 

9-401569 

^^^^1 

832 

692224 

675030368 

28-8444102 

9-405338 

^^ 

833 

693880 

578009537 

28-8017394 

9-409105 

834 

095556 

580093704 

28-8790582 

9-412869 

II 

B3& 

697-225 

582182876 

28-8963660 

9-410030 

b 

836 

608896 

684277056 

28-9130640 

9-420387 

■ 

837 

700569 

586376253 

28-9309523 

9-424142 

■ 

838 

702244 

588480472 

28-9482297 

9-427893 

■ 

830 

703021 

5905*19719 

28-9654967 

9-431642 

F 

840 

705800 

592704000 

289B27S35 

9-435388 

841 

707281 

594823321 

290000000 

9-439130 

842 

708904 

696947688 

29-0172303 

9-442870 

wa 

710<i49 

509077107 

29-0344623 

9-446007 

844 

712336 

6012 11584 

20-0516781 

^^^^1 

846 

714026 

603351126 

29-0688837 

9-454071 

^^^^1 

846 

715716 

605495736 

29-0860791 

9-457799 

^^^^1 

847 

717409 

607045423 

20-1032644 

9-461534 

848 

719104 

609800192 

29-1204396 

9-466247 

■ 

849 

720801 

611960049 

29- 1378016 

9-468966 

1 

860 

722500 

614125000 

291647595 

0-472082 

1 

8S1 

724201 

616295061 

29-1719043 

9-476396 

■ 

862 

726904 

018470208 

20-1890390 

9-480100 

1 

663 

727600 

620630477 

20-2061637 

9-483813 

1 

HM 

720316 

622835864 

20-2232784 

9-487618 

J 

856 

731026 

625026375 

29-2403830 

9-491219 

866 

732736 

627222016 

29-2574777 

9-484918 

^^^^1 

867 

734449 

629422793 

20-2745623 

9-498614 

^^^^M 

868 

736164 

031628712 

29-2916370 

9-502307 

^^^^M 

661) 

737881 

633839770 

29-3087018 

9-505908 

^^^^M 

860 

730«(K) 

636056000 

203'267566 

9-609085 

^^^^M 

801 

741321 

630277381 

29-34-28013 

9-613369 

^^^^M 

802 

743044 

640603928 

20-3698363 

9-517051 

8S3 

744709 

642735047 

29-3768616 

0-6207af 

^^^H 

Mi4 

746496 

044972544 

29-3938769 

9-624400 

1 

866 

748225 

647214625 

29-4108823 

9-628079 

H 

^^ 

Ban 

749956 

649461896 

29-4278779 

9-531749 

H 

B 

HOT 

7&1680 

031714363 

29-4448637 

0-536417 

1 

1 

1 

■ 

U 

390 


Numbo*. 

Square. 

Cube. 

Square  Root 

Cube  RooL 

808 

763424 

663072032 

20-4618307 

9-639081 

860 

766161 

666234000 

20-4788060 

9-642748 

870 

766000 

668603000 

20-4067624 

9-646402 

871 

768641 

660776311 

20-6127001 

9-660068 

872 

760384 

663064848 

20-6296461 

9-663713 

873 

762120 

666338617 

20-6466734 

9-667368 

874 

763876 

667627624 

20-6634910 

9-661010 

876 

766626 

660021876 

20*6803989 

9-664666 

876 

767376 

672221376 

29-5972972 

9-668288 

877 

760120 

674626133 

29-6141868 

9-671937 

878 

770884 

676836162 

20-6310648 

9-676574 

870 

772641 

670161430 

20-6470342 

9-67920B 

880 

774400 

681472000 

20-6647030 

9-682830 

881 

776161 

683707841 

20-6816442 

,9-686468 

882 

777024 

686128068 

20-6084848 

9-690098 

883 

770680 

688466387 

20-7163160 

0-603716 

884 

781466 

600807104 

20-7321376 

9-697387 

885 

783226 

603164 L26 

20-7480406 

9-600964 

886 

784006 

606606466 

20-7667621 

9-604660 

887 

786760 

607864103 

20-7826462 

9-608181 

888 

788644 

700227072 

29-7003280 

9-611791 

880 

700321 

702606360 

20-8161030 

9-616397 

800 

702100 

704060000 

20-8328678 

9-619001 

801 

703881 

707347071 

29*8496231 

9-622608 

802 

706664 

700732288 

29-8663690 

9-626201 

803 

707440 

712121067 

20*8831066 

9-629797 

804 

700236 

714616084 

20-8008328 

9-633390 

806 

801026 

716017376 

20-0165606 

9-636961 

H06 

802816 

7103231:KJ 

200332501 

9-640669 

807 

804600 

721734273 

200400583 

9-644164 

808 

806404 

724160702 

200600481 

9-647736 

800 

808201 

720572000 

200833287 

9-661316 

000 

810000 

720000000 

30-0000000 

9-664898 

001 

811iU)l 

731432701 

30-0100020 

9-668468 

002 

813604 

733870808 

30-0333148 

9-662040 

003 

816400 

730314327 

30-0400584 

9-666609 

004 

817216 

738703204 

30-0005028 

0-669176 

906 

810026 

741217026 

30-0832170 

9-672740 

006 

820836 

743077410 

30-0008330 

9-676801 

007 

822040 

740142043 

30-1104407 

9-679860 

008 

824404 

748013312 

30-1330383 

9-688416 

000 

820281 

761080420 

301490200 

9-686970 

010 

828100 

763571000 

30-166-2063 

9-690631 

Oil 

820021 

760068031 

30-1827766 

9-694009 

012 

831744 

768660628 

30-1993377 

9-697616 

Old 

833600 

701048407 

30-2168809 

9-701168 

014 

836300 

703661044 

30-2324329 

9-704608 

016 

837226 

706000876 

30-2480669 

9-708286 

016 

830060 

768676206 

30-2664919 

9-711773 

017 

840880 

771006213 

30-2820079 

9^16806 

018 

842724 

773620632 

30-2086148 

8^18886 

1 

m 

m 

^ 

^ 

1 

Nmnber. 

Square. 

Cube. 

Square  Root 

CuW  RooL 

919 

844361 

776151550 

30-3130128 

0-722383 

I 

920 

846400 

778088000 

30-3315018 

9-725888 

■ 

921 

848241 

781220061 

30-3470818 

9-729410 

■ 

922 

850084 

7H3777448 

30-3644529 

0-732930 

■ 

923 

851020 

786330407 

30-38091 61 

9-730448 

■ 

924 

853776 

78888)1024 

30-3973083 

9-739903 

■ 

925 

865C25 

701453125 

30-4138127 

9-743476 

w 

92fl 

B57476 

704022776 

30-4302481 

9-746085 

927 

859320 

700507083 

30-4466747 

0-750493 

928 

801184 

700178752 

!M)-4e30024 

9-753998 

928 

863041 

801765089 

30-4795013 

9-757500 

930 

804900 

804357000 

30-4050014 

9-76  lOOO 

931 

800701 

800954491 

30-5122920 

9-764497 

932 

808024 

B00557568 

30-5286750 

9-767992 

933 

870489 

812160237 

30-5450487 

0-771484 

934 

872350 

814780504 

30-5614136 

9-774974 

936 

874225 

817400375 

30-5777097 

9-778461 

93G 

876006 

820025850 

30-5041171 

9-7«2946 

r 

037 

877060 

822656953 

30-6104557 

9785428 

M 

938 

870844 

825203672 

30-6267857 

9-788908 

■ 

039 

881721 

827036010 

30-0431069 

9-792380 

■ 

940 

883600 

830584000 

30-6504194 

9-795861 

941 

8854B1 

833237621 

30'6767233 

9-799333 

M2 

887304 

835896888 

30-6930185 

9-802803 

043 

880249 

83850 IB07 

30-7083051 

9-806271 

044 

801136 

8412323(14 

30-7245830 

9-809736 

045 

803026 

8439(18625 

30-7408523 

9-813198 

ft 

840 

804916 

846590536 

30-7571130 

9-816659 

■ 

047 

806800 

B49278123 

30-7733651 

9020117 

■ 

»4S 

898704 

851071302 

30-7896086 

0-823572 

■ 

04» 

900({01 

854070310 

30-8058430 

0-827026 

■ 

950 

902500 

857375000 

30-8220700 

9-830475 

■ 

951 

904401 

860085351 

30-8382079 

9-833923 

w 

9S4 

900304 

062801408 

30-8544972 

9-837360 

053 

008200 

865523177 

30-8700901 

9-840812 

954 

010116 

068250004 

30-8808904 

9-844253 

055 

012025 

870083875 

300030743 

9847692 

' 

056 

013030 

873722810 

30-0192497 

9-851120 

957 

015849 

B70467493 

30-9354166 

9-854501 

958 

017764 

870217912 

300515751 

0-857902 

959 

919081 

881074070 

30-9077251 

9-801421 

900 

921600 

884730000 

30-0838608 

0-864848 

981 

923521 

887503681 

310000000 

0-868272 

002 

925444 

800277128 

310101248 

9-871694 

903 

927309 

003050347 

31-0322413 

98751 13 

964 

02i»29e 

80584  i:f44 

31-0403404 

9-878530 

905 

931225 

808632125 

31-0644491 

9-881945 

966 

033150 

901428006 

31-0805405 

0-886337 

967 

035089 

004231063 

31-0066-230 

9-888767 

908 

937024 

907030232 

31-1I26084 

9-892174 

069 

938961 

909853209 

31-1287048 

9-895580 

392 


Number. 

Square. 

Cube. 

Square  Root. 

Cube  Root 

970 

940900 

912673000 

311 448230 

9-898983 

971 

942841 

915498611 

31 

1608729 

9*902383 

972 

944784 

918330048 

31 

1769145 

9*905781 

973 

946729 

921167317 

31- 

1929479 

9-909177 

974 

948676 

924010424 

31' 

2089731 

9-912571 

976 

950625 

926859375 

31 

2249900 

9-915962 

976 

952576 

929714176 

31 

2409987 

9-919351 

977 

954529 

932574833 

31 

-2569992 

9-922738 

978 

956484 

935441352 

31 

2729915 

9-926122 

979 

958441 

938313739 

31 

►2889757 

9-929504 

980 

960400 

941192000 

31 

•3049517 

0-932883 

981 

962361 

944076141 

31 

3209195 

9-936261 

982 

964324 

946966168 

31 

3368792 

9*939636 

983 

966289 

949862087 

31 

3528308 

9-943009 

984 

968256 

952763904 

31* 

3687743 

9*946379 

985 

970225 

955671625 

31 

3847097 

9-949747 

986 

972196 

958585256 

31' 

4006369 

9-953113 

987 

974169 

961504803 

31 

4165561 

0-956477 

988 

976144 

964430272 

31' 

4324673 

9-959839 

989 

978121 

967361669 

31' 

4483704 

9-963198 

990 

980100 

970299000 

31' 

4642654 

0-966554 

991 

982081 

973242271 

31 

4801525 

9-969909 

992 

984064 

976191488 

31 

'4960315 

9-973262 

993 

986049 

979146657 

31 

5119025 

0-076612 

994 

988036 

982107784 

31- 

5277655 

0-070050 

995 

990025 

985074875 

31 

5436206 

0-083305 

996 

992016 

988047936 

31 

5594677 

0-986648 

997 

994009 

991026973 

31 

5753068 

9-989990 

998 

996004 

994011992 

31 

5911380 

9-993328 

999 

998001 

997002999 

31*6069613 

9-996665 

*^*  In  this  edition  some  errors  in  the  table  of  Dr.  Hntton  have  been 
corrected.  The  most  extensive  and  accurate  tables  of  the  powers  of  num- 
bers, are  those  published  by  Professor  Barlow,  of  the  Royal  Military 
Academy,  entitled  New  Mathematical  Tables^  which  also  contain  tables  of 
prime  numbers  and  factors,  so  useful  in  arranging  the  numbers  of  wheel 
work. 


APPENDIX   B. 


IRODUCTION  OF   THE   SLIDE  PRINCIPLE 


TOOLS   AND   MACHINES  EMPLOYED   IN   THE 
PRODUCTION  OP  MACHINERY. 


BY  JAMES  NASMYTH. 


The  striking  and  rapid  progress  which  has  within  the 
last  thirty  years  taken  place  in  the  perfection  of  all  descrip- 
tions of  machinery,  not  only  as  regards  a  more  complete  and 
sound  knowledge  of  the  principles  of  mechanical  or  con- 
structive science,  as  exhibited  in  the  general  arrangement 
of  the  parts,  but  more  especially  in  respect  to  the  increased 
perfection  of  the  workmanship,  which  is  now  so  generally 
met  with  in  the  vast  variety  of  machines  which  are  yearly 
sent  forth,  as  it  were  to  proclaim  new  triumphs  over  mat- 
ter, cannot  but  lead  us  to  endeavour  to  find  a  cause  for 
so  remarkable  and  important  a  feature  in  the  history  of 
mechanism. 

481.  In  pursuing  this  inquiry,  we  shall  find  that  the  accu- 
mulated experience  and  skill  in  constructive  science,  which 
has  resulted  from  a  continually  increasing  demand  for  ma- 
chinery, will  only  throw  light  on  one  portion  of  this  in- 
teresting subject ;  inasmuch  as  increased  experience  alone 
ill  not  sufficiently  account  for  the  almost  mathematical 


394  NASMTTH    ON    TOOLS 

accuracy  and  precision  which  we  find  existing  in,  and  con- 
ferred on,  the  forms  of  the  various  details,  whether  of  the 
most  delicate  or  ponderous  machines ;  to  have  produced 
which,  were  it  even  possible  by  manual  dexterity  and 
labour,  would  have  entailed  so  vast  an  expense  in  construc- 
tion, that  neither  in  respect  to  quantity  or  price  could  we 
have  ever  hoped  to  be  able  (even  with  our  present 
mechanical  population  increased  tenfold)  to  have  kept  pace 
with  the  demand  which  has  resulted  from  the  increased 
perfection  and  facilities  of  production  realized  by  improved 
mechanism. 

482.  Viewing  abstractedly  the  forms  of  the  various  details 
of  which  every  machine  is  composed^  we  shall  find  that  they 
consist  of  certain  combinations  of  six  primitive  or  ele- 
mentary geometrical  figures,  namely,  the  line,  the  plane, 
the  circle,  the  a/Under,  the  cone,  and  the  sphere  ;  and  that, 
however  complex  the  arrangement,  and  vast  the  number 
of  the  parts  of  which  a  machine  consists,  we  shall  find 
that  all  may  be  as  it  were  decomposed  and  classed  under 
these  six  forms ;  and  that,  in  short,  every  machine,  what- 
ever be  its  purpose,  simply  consists  of  a  combination  of 
these  forms,  more  or  less  complex,  for  the  attainment  of 
certain  objects  and  performance  of  required  duties.  It 
therefore  follows,  that  the  more  near  to  absolute  mathema- 
tical truth  we  can  have  the  forms  of  those  parts,  the  more 
perfectly  will  the  machine  perform  its  duties. 

483.  Up  to  within  the  last  thirty  years,  nearly  every  part 
of  a  machine  had  to  be  made  and  finished  to  its  required 
form,  by  mere  manual  labour ;  that  is,  on  the  dexterity  of 
the  hand  of  the  workman,  and  the  correctness  of  his  a/e, 
had  we  entirely  to  depend  for  accuracy  and  precision  in 
the  execution  of  such  machinery  as  was  then  required; 
consequently,  the  enormous  expense  which  was  incurred  in 
such  attempts,  even  in  the  production  of  comparatively 
simple  machines,  in  most  cases  proved  a  fonnidable  barrier 


AND    MACHINES.  395 

to  the  supply  of  such  as  the  increasing  wants  of  civiliz- 
ation rendered  desirable,  and  when  at  length  the  success- 
ful efforts  of  Watt  and  Arkw-right  produced  such  an 
entire  revolution  in  the  steam  engine  and  cotton  manu- 
facture, and  so  disclosed  to  mankind  such  vast  mines  of 
wealth  in  the  latent  powers  of  production,  and  capabilities 
of  every  country,  and  as  the  only  obstacle  to  the  attainment 
of  so  desirable  an  end  consisted  in  our  almost  entire  de- 
pendence upon  manual  dexterity  for  the  formation  and  pro- 
duction of  such  machines  as  were  required,  the  necessity 
of  more  trustworthy  and  productive  agents  rendered  some 
change  in  the  system  imperative.  In  short,  a  sudden  de- 
mand for  machinery  of  unwonted  accuracy  arose,  while  the 
stock  of  workmen  then  existing  were  neither  adequate  in 
respect  to  number  or  ability  to  meet  the  wants  of  the  time, 
and  but  for  the  introduction  of  the  principle  which  I  am 
about  to  describe,  we  never  could  hav(!  attained  to  one- 
thousandth  part  of  the  bright  objects  which  were  then  dis- 
closed to  view,  and  which  have  since  been  so  wonderfully 
Bid  amply  realized. 
484.  The  principle  to  which  I  allude  consists  in  the  sub- 
'  stitution  of  a  mechanical  contrivance  in  place  of  the  human 
hand,  for  holding,  applying,  and  directing  the  motions  of  a 
cutting  tool  to  the  surface  of  the  work  to  be  cut,  by  which 
we  are  enabled  to  constrain  the  tool  to  move  along  or  across 
the  surface  of  the  object  with  such  absolute  precision, 
that  with  scarce  any  expenditure  of  force,  and  indeed,  in 
most  cases,  none  at  all  on  the  part  of  the  workman,  (as 
^ahall  be  seen  presently,)  we  are  enabled  to  produce  any 
^kf  the  before-named  clementar)'  geometrical  forms  with 
^K  degree  of  accuracy,  case,  and  rapidity,  as  compared  with 
^Hie  old,  imperfect,  hand  system,  as  may  well  be  considered 
Hp  mightj-  triumph  over  matter ;  and  the  more  justly  so,  ' 
^^hen  we  behold  the  vast  results  which  improved  machi- 
nery is  enabling  us  to  bring  about,  ali  of  which  may,  in  a 


etosi 

fri 


39G 


NASMYTH    ON   TOOLS 


more  or  less  direct  manner,  be  traced  back  to  the 
of  power  which  we  have  acquired  hy  means  of  the 
introduction  and  appUcation  of  the  slide  rest  principle. 


AND  MAcnmEs.  397 

485.  How  it.  has  liappenetl  that  the  inestimable  merits  of 
lis  contrivance  have  not  been  more  justly  appreciated,  and 
as  it  were  unobserved,  it  is  difficult  to  account ;  it 
ly  be  that  its  beautiful  simplicity  has  been  overlooked  in 
the  glare  of  dazzling  results  which  it  has  produced ;  it  is 
only  by  considering  how  we  could  "get  on"  without  its  im- 
portant help,  that  the  real  value  of  this  admirable  con- 
trivance appears  before  us  in  its  true  light. 

48(j.  It  is  not  indeed  saying  at  all  too  much  to  state,  that 
its  influence  in  improving  and  so  extending  the  use  of  ma- 
linerj-,  has  been  as  great  as  that  produced  by  the  im- 
ivemcnt  of  the  steam  engine  in  respect  to  perfecting 
lufactures  and  extending  commerce,  inasmuch  as  with- 
it  the  aid  of  the  vast  accession  to  our  power  of  producing 
perfect  mechanism,  which  it  at  once  supplied,  we  could 
never  have  worked  out  into  practical  and  profitable  forms 
the  conceptions  of  those  master-minds  who,  during  the  last 
centurj,  have  so  successfully  pioneered  the  way  for 
ikind  ever  after  attaining  the  otherwise  latent  trea- 
sures of  the  material  world,  even  although  opposed  by 
time,  space,  and  the  elemental  1  regret  much  that  my 
limits  will  not  permit  me  to  trace  in  detail,  through  all 
their  ramifications,  the  almost  infinite  benefits  which  have 
been  conferred  on  mankind  by  our  having  (through  means 
of  this  admirable  slide  rest  principle)  obtained  a  most 
complete  and  signal  triumph  over  the  material  world. 
steam  engine  itself,  which  supplies  us  with  such  un- 
ided  power,  owes  its  present  perfection  to  this  ad- 
'able  means  of  giving  to  metallic  objects  the  most  precise 
and  perfect  geometrical  forms.  How  could  we,  for  in- 
stance, have  good  steam  engines,  if  we  had  not  the  means 
<rf  boring  out  a  true  cylinder,  or  turning  a  true  piston  rod,  or 
ing  a  valve  face?  It  is  this  alone  which  hius  fiiniished 
with  the  means  of  carrying  into  ])riictice  the  accumulated 
of  scientific  investigation  in  mechanical  subjects, 
u  D 


me  ( 


^^pral 


398  NASMYTH    ON    TOOLS 

487.  With  a  view  to  render  the  preceding  remarks 
more  generally  understood,  I  have  given  the  annexed  sketch, 
in  order  to  illustrate  the  advantages  of  the  slide  rest  prin- 
ciple, as  a  substitute  for  manual  labour  and  dexterity,  in  the 
case  of  the  turning  lathe  ;  the  more  so,  as  it  was  in  this 
form  and  application  in  which  its  admu*able  merits  became 
first  known  to  the  mechanical  world. 

488.  Fig.  1.  represents  the  system  of  hand  turning  in 
general  practice  previous  to  the  introduction  of  the  slide  rest 
Here  it  will  be  seen  that  the  workman  has  no  other  means 
of  applying  and  guiding  his  tool  to  the  work  in  the  lathe, 
than  his  mere  unaided  muscular  strength,  the  expenditure 
of  which,  in  the  case  of  turning  large  objects,  would  be  so 
great,  that  he  could  stand  it  for  no  length  of  time ;  and 
even  if  he  were  able,  he  would  have  to  depend  on  his 
strength  and  dexterity  alone  for  producing  even  the  hum- 
blest and  most  plain  class  of  work.  By  such  means  of  this 
nature  as  were  generally  practised  before  the  introducticm 
of  the  slide  rest,  we  could  only  attain  to  any  thing  hke 
true  work,  by  an  almost  infinite  expenditure  of  labour,  for 
with  the  utmost  care  on  his  part  he  could  not  avoid  occa- 
sionally cutting  a  little  too  deep,  the  consequence  of  which 
would  be  that  he  would  require  to  go  all  over  the  rest  of 
the  surface,  in  order  as  it  were  to  lower  it  to  the  level  of 
the  accidentally  too  deep  cufe  just  named ;  in  most  cases,  in 
so  doing,  he  would  make  the  work  too  small,  or  have  occa- 
sion either  to  leave  the  mark  in  the  bar,  or  else  alter  all 
his  measures  to  suit  the  bad  results  of  depending  on  the 
chance  of  his  dexterity.  It  will  be  seen  that  the  workman 
in  Fig.  1.  rests,  or  obtains  support  for,  the  end  of  his  tool, 
so  as  to  resist  the  force  of  the  cut,  by  placing  it  upon  "  the 
rest'*  R. 

489.  Now  let  us  just  suppose  that  instead  of  holding  his 
tool  with  his  hands,  that  he  had  it  bolj:ed  firm  to  this  same 
rest,  and  that  while  it  was  cutting  a  shaving  from  the  bar 


I 


tf 


AND    MACHINES.  399 

lathe,  that  lie  had  means  of  sliding  the  rest  with  its 
tool  along  the  bed  of  the  lathe,  parallel  to  the  axis  of  the 
work,  it  is  evident  that,  in  so  doing,  we  should  be  able  to 
turn  the  bar  quite  true ;  and  if  a  screw  was  provided  for 
the  purpose  of  giving  this  sliding  motion,  we  should  then 
have  a  slide  rest ;  exactly  in  such  manner  was  this  truly 
admirable  tool  introduced  to  the  mechanical  world.  On 
reference  to  Fig.  2,  it  will  be  at  once  seen  that  these  ob- 
jects  are  attained  in  a  very  simple  manner.  The  tool  is  in 
this  case  held  fast  and  firm  by  a  species  of  iron  hand  or 
"rice,  while  it  is  constrained  to  move  in  a  definite  direction 
by  means  of  the  slide  s,  (see  Fig.  3,)  the  sliding  motion 
l)eing  communicated  by  the  hand  of  the  workman  to  the 
screw  handle  h,  the  required  depth  of  cut  being  regulated 
by  the  under  slide  k,  operated  upon  in  like  manner  by  a 
screw  and  handle ;  so  that  by  the  separate  or  comhined 
motion  of  these  two  slides,  the  point  of  the  tool  can  be 
made  to  traverse  along  or  across  the  work  as  required,  with 
an  expenditure  of  power  on  the  part  of  the  workman  so 
trifling  as  scarce  to  be  appreciated  ;  and  with  such  a  degree 
of  definite  and  precise  accuracy  will  the  tool  by  these  means 
move,  that,  after  setting  the  tool  to  work,  he  needs  not  to  look 
at  it  80  long  as  he  simply  keeps  turning  the  screw  handle  j 
and  by  a  very  simple  contrivance,  which  we  have  en- 
deavoured to  exhibit  in  Fig.  3,  x,  the  attendance  of  the 
workman  ia  entirely  dispensed  with  by  the  introduction  of 
le  self-acting  principle,  by  which  the  revolution  of  the 
irk  in  the  lathe  is  made  to  supply  the  place  of  the  hand 
the  workman.  As  may  be  seen  at  x,  Fig.  3,  by  simply 
fixing  to  the  work  in  the  lathe  a  piece  of  iron  as  at  o,  and 
placing  on  the  end  of  the  screw  s  of  the  upper  slide  a  star 
wheel  X,  it  is  evident  that  at  each  revolution  of  the  work 
in  the  lathe  the  end  of  the  iron  finger  x  will  come  in  con- 
tact with  one  of  the  teeth  of  the  wheel,  and  move  it  round 
at  each  turn,  bringing  the  next  in  succession  into  a 
D  n  2 


400 


NASMYTH   ON   TOOLS 


situation  so  as  in  like  manner,  at  each  revoluticm  of  the 
work,  the  screw  wheel  x  is  moved  round,  and  the  tool  by 
that  simple  means  slid  by  successiye  steps  along  the  sur£ace 
of  the  work ;  here,  then,  by  this  simple  adaptation,  we 
have  not  only  done  away  with  necessity  for  a  dexterous 
workman,  but  have  entirely  removed  all  necessity  of  at- 
tendance whatsoever  during  the  progress  of  the  tool 
over  the  slide  length  of  the  surface  of  the  work. 


'Tu-- 


^    -A- 


This  will  in  some  degree  convey  an  idea  of  the  nature  of 
the  self-acting  principle,  by  the  adoption  of  which  we  are 
enabled  to  elevate  to  so  high  a  degree  the  productive 


I 
I 


■     mi 

K' 


AND    MACHINES.  401 

powers  of  our  workmen  and  machinery.  There  are  a  vast 
variety  of  modes  of  attaining  this  self-acting  motion,  but 
the  one  above  alluded  to  will  be  sufficient,  the  more  so  as 
it  is  the  most  generally  employed,  and  most  simple. 

490.  It  was  this  holding  of  a  tool  bymeans  of  an  iron  hand, 
and  constraining  it  to  move  along  the  surface  of  the  work 
in  so  certain  a  manner,  and  with  such  definite  and  precise 
motion,  which  formed  the  great  era  in  the  history  of  me- 
chanism, inasmuch  as  we  thenceforward  became  pos- 
Besscd,  by  its  means,  of  the  power  of  operating  alike  on 
the  most  ponderous  or  delicate  pieces  of  machincrj'  with  a 
degree  of  minute  precision,  of  which  language  cannot  convey 
an  adequate  idea  j  and  in  many  cases  we  have,  through 
its  agency,  equal  facility  in  carrying  on  the  most  perfect 
workmanship  iu  the  interior  parts  of  certain  machines, 
where  neither  the  hand  nor  eye  can  reach,  and  nevertheless 
we  can  give  to  these  parts  their  reqiiired  form  with  a  de- 
of  accuracy  as  if  we  had  the  power  of  trajistbrming 
Ourselves  into  piginy  workmen,  and  so  apply  our  labour  to 
the  innermost  holes  and  comers  of  our  machinery. 

4>91'  It  would  be  blamable  indeed  (after  having  en- 
deavoured  to  set  forth  the  vast  advantages  which  have 
been  conferred  on  the  mechanical  world,  and  therefore  on 
mankind  generally,  by  the  invention  and  introduction  of  the 
slide  rest)  were  I  to  suppress  the  name  of  that  admirable  in- 
diridual  to  whom  we  are  indebted  for  this  powerful  agent 
towards  the  attainment  of  mechanical  perfection.  I  allude  to 
the  late  Henry  Maudslay,  engineer,  of  London,  whose  useful 
life  was  enthusiastically  devoted  to  the  grand  object  of  im- 
proving our  means  of  producing  perfect  workmanship  and 
machinery ;  to  him  we  are  certainly  indebted  for  t/ie  slide 
fsi,  and  consequently,  to  say  the  least,  we  are  indirectly 
for  the  vast  benefits  which  have  resulted  from  the  intro- 
duction of  so  powerful  an  agent  in  perfecting  our  ma- 
chinery and  mechanism  generally.     The  indefatigable  care 


402  NASMYTH    ON    TOOLS 

which  he  took  in  inculcating  and  diffiising  among 
workmen,  and  mechanical  men  generally,  sound  ideas  of 
practical  knowledge  and  refined  views  of  construction,  has 
rendered  and  ever  will  continue  to  render  his  name  iden- 
tified with  all  that  is  nohle  in  the  amhition  of  a  lover  of  me- 
chanical perfection.  The  vast  results  which  have  sprung  firom 
his  admirable  mind,  is  his  best  monument  and  eulogium. 

492.  The  vast  practical  advantage  which  resulted  from 
the  substitution  of  **  the  slide  rest ''  in  place  of  the  hand 
in  the  process  of  turning,  had  its  natural  efifect  in  causing 
its  adoption  and  application  to  other  important  processes 
in  constructive  science.  So  striking  and  certain  were  the 
effects  and  advantages  as  respects  the  superior  quality  and 
cheapness  of  the  work  produced  by  its  means,  that  it  soon 
induced  a  very  marked  change  in  mechanical  designs,  in- 
asmuch as  this,  that  many  improved  arrangements  in  me- 
chanism had  been  kept  back  from  the  vast  expense  attend- 
ant on  the  employment  of  certain  forms  in  the  parts,  such 
as  perfectly  true  cylindrical  rods  or  circular  or  flat  surfaces, 
which  the  important  aid  of  the  slide  rest  now  renders  so 
c^heap,  (comparatively  speaking,)  that  every  practical  en- 
gineer, in  making  out  his  design  in  detail,  had  only  to  keep 
in  mind  the  vast  capabilities  and  powers  of  the  slide  rest, 
to  enable  his  fancy  to  luxuriate  in  the  introduction  of  the 
most  perfect  geometrical  forms,  as  not  only  attainable  in 
practice,  but  actually  the  cheapest  forms  through  whose 
agency  he  could  attain  his  object.  I  have  every  reason,  in- 
deed, to  call  the  introduction  of  the  slide  rest  a  great  era 
in  the  history  of  mechanism,  as  every  piece  of  machinery 
which  was  produced  by  its  agency,  bore  such  evident  marks 
of  superiority,  as  very  rapidly  and  extensively  proclaimed 
to  the  mechanical  world  that  a  great  step  (leap  forward,  I 
should  rather  say)  had  been  made,  and  in  proof  of  it,  we 
have  only  to  look  around  us  at  this  day  to  see  what  is  doing 
by  improved  machinery,  to  place  beyond  doubt  what  I  have 


AND    MACHINES.  403 

i  as  to  this  era  in  mechanism — "  the  introduction  of 
the  slide  rest." 

493.  Were  I  to  attempt  to  trace  in  detail  tho  almost  in- 
finite application  of  the  slide  rest  principle,  I  should  re- 
quire to  describe  almost  every  machine  which  is  employed  in 
giving  definite  forms  to  materials ;  but  as  such  would  he  in- 
compatible with  my  limita,  I  shall  confine  myself  to  one  or 
two  of  tho  more  generally  used  and  important  applications ; 
and  in  endeavouring  to  do  so,  I  shall,  for  the  sake  of  clear- 
ness, avoid  those  minute  details  which,  although  most  fre- 
quently combined  with  the  slide  principle,  yet  are  so  sub- 
ordinate, and  so  frequently  varied  according  to  the  taste  of 

tthc  engineer,  that  it  is  best  to  strip  them  from  the  simple 
ilustrations  I  have  endeavoured  to  give,  so  as  to  leave,  as 
t  were,  more  prominent  and  conspicuous  iJte  principle  of 
[he  machine. 

494.  I  cannot  properly  introduce  to  the  attention  of  my 
peaders  a  more  worthy  and  truly  important  immediate  de- 

iendant  of  "  the  slide  rest "  than  the  plmii/ig  macluTie, 
which  has  done  more  within  the  last  10  or  15  years  for 
reducing  the  cost,  and  for  extending  the  use  of  perfect  ma- 
chinery, than  had  been  the  case  by  all  the  improvements 

1  mechanism  for  the  last  century. 
¥J5.  There  is  no  form  which  is  so  frequently  reijuired 
and  essential  to  any  piece  of  mechanism  as  the  plane  sur- 
fece,  or  rectangular  prismatic  forms  generally, 

496.  The  vast  expense  attendant  on  the  production  of 
such,  by  the  tedious  and  unsatisfactory  process  of  chipping 
and  filing,  caused  every  engineer  to  avoid  by  all  means  any 
arrangements  which  rendered  such  forms  necessary,  how- 
ever essential  they  might  be  to  the  perfect  action  of  the 
machine.  It  is  quite  laughable  to  observe,  in  any  old  piece 
of  mechanism,  the  niggardly  use  of  those  important  forms 

ffr*im  the  above  obstacle.     The  introduction  of  the 
machine  at  once  altered  the  entire  system,  inas- 


rea 
^^chi] 

V     and 


404  NASMYTH    ON    TOOLS 

much  as  forms  and  arrangements  became  practically  pos- 
sible,  which  formerly  the  engineer  dared  not  think  of  using. 
This  was  simply  following  out  in  the  plane  surface,  what 
the  slide  rest  had  produced  in  the  turning  lathe  as  regards 
cylindrical  forms ;  and  the  result  was,  that  not  only  was 
the  machinery  produced  by  its  agency  most  strikingly  su- 
perior, by  its  direct  influence,  but  also  as  the  planing  ma- 
chine enabled  us  to  produce  improved  tools  at  so  very  much 
reduced  cost,  that  mighty  principle  in  all  affairs,  (namely, 
cause  and  effect  tearing  each  other  alternately.)  The  first 
planing  machine  enabled  us  to  produce  the  second  still 
better ;  that  again  produced  a  better  still ;  and  now  shde 
rests  of  the  most  perfect  kind  came  streaming  forth  from 
them,  and  they,  again,  assisted  in  making  better  still ;  so 
that  in  a  very  short  time  a  most  important  branch  of  en- 
gineering business,  namely,  tool-making,  arose,  which  had 
its  existence  not  merely  owing  to  the  demand  preexisting 
for  such  improved  tools,  but  in  fact,  raised  upon  a  de- 
mand as  it  were  of  its  own  creating,  and  all  this  caused  by 
the  slide  rest,  and  its  offspring,  the  planing  machine.  One 
has  only  to  go  into  any  of  those  vast  establishments,  which 
within  the  last  10  years  have  sprung  up  for  the  purpose  of 
supplying  the  demand  for  machinery,  and  we  shall  find 
that  nine-tenths  of  all  the  fine  mechanism  in  use,  and  in 
process  of  production,  is  through  the  agency,  more  or  less 
direct,  of  the  slide  rest  and  planing  machine. 

497-  Figure  4  represents  the  general  arrangement  of 
parts  existing  in  most  planing  machines.  It  consists  of 
two  principal  parts,  namely,  the  bed  b  on  which  the  table 
T  slides  by  certain  mechanism  backward  and  forward,  so 
that  any  piece  of  work,  w,  being  bolted  to  it,  partakes  of 
the  same,  as  if  it  were  a  part  of  the  table  t  ;  the  table  t 
being  constrained  to  move  in  a  perfectly  straight  line  to 
and  fro,  by  its  sliding  on  the  two  angular  ridges,  c  c. 


AND   MACHINES. 


/%|A^ 


498.  Over  the  table  T  is  fixed  "  a  slide  rest "  s,  which 
is  held  fast  by  being  bolted  to  the  two  upright  standarda 
N  N.  This  slide  s  has  a  transverse  slide  d,  which  serves  to 
hold  the  tool  in  such  a  manner  that  it  may  be  lowered  down 
and  adjusted  so  as  to  cause  the  tool  to  take  a  cut  more  or 


406  NASMYTH    ON   TOOLS 

less  deep  as  desired,  which  adjustment  is  performed  hy  the 
handle  l,  so  that  every  time  the  tahle  and  the  work  fixed 
to  it  moves  to  and  fro,  the  tool  in  the  down  slide  d,  is  by 
certain  apparatus  moved  each  time  a  little  way  across  the 
table,  so  that  by  a  repeated  series  of  sliding  backwards 
and  forwards  of  the  table,  the  tool  is  made  to  traverse  the 
surface  of  the  work,  and  in  so  doing  it  transfers  the  per- 
fectly true  figure  of  the  slide  s,  on  to  that  of  the  surface 
of  the  work  w,  and  so  produces  a  perfect  plane  surface.  I 
trust  an  inspection  of  the  figure  will  do  more  to  render  this 
clear,  than  any  further  attempt  at  description. 

499*  As  to  the  means  of  giving  motion  to  the  table,  as 
also  to  the  screw  of  the  slide  s,  it  is  not  required  here  to 
enter  into  such  details,  as  they  vary  so  much  according  to 
the  fancy  of  different  makers,  who  have  each  their  peculiar 
fancy  as  to  the  best  arrangement. 

500.  An  inspection  of  the  figiure  will,  I  trust,  satisfy  any 
one  that  this  machine  is  derived  from  the  slide  rest,  for 
the  slide  s  is  nothing  more  than  a  slide  rest,  held  to  its 
work  by  the  two  standards  nn,  while  the  work  w  repre- 
sents a  surface  on  the  lathe,  which  is  made  to  move  in  a 
straight  line,  in  place  of  a  revolving  motion,  as  it  would 
have  done  had  it  been  a  cylindrical  surface  being  turned 
in  the  lathe.  This,  indeed,  is  my  main  object  in  giving 
this  figure,  as  it  serves  to  show  that  it  is  to  the  slide  rest 
system  that  we  are  indebted  for  the  planing  machine,  how- 
ever  varied  the  constru^ve  details  of  such  planing  ma- 
chines as  we  mejt  y^idx  may  be,  yet  we  shall  find  that  they 
all  embody  the  above  principal  arrangements,  and  are  all 
slide  rests  for  turning,  i.  e.  planing^a^  work. 

501.  Again,  in  the  case  of  the  screw-cutting  machine, 
we  shall  find  (Fig.  5)  that  it  consists  simply  of  a  slide  re^t, 
which  receives  its  sliding  motion  from  the  revolution  of  the 
spindle  or  work  in  the  lathe.  I  have  chosen  the  latter,  as 
it  tends  to  render  the  arrangement  more  distinct. 


AND   MACHINES. 


407 


Here  we  have  the  slide  rnrt  s,  whose  tool-holder  is  slid  along 
1^  means,  of  .the  screw  b>  which  receives  its  motion  from  the 
work  in  the  lathe>  by  means  of  the  wheels  w  w,  by  which 
it  is  evident,  that  as  the  work  x  revolves  in  the  lathe,  a 
revolving  motion  will  be  transferred  to  the  screw  s,  and  the 
pmnt  of  the  tool  will»  ob  sliding  almg,  have  a  spiral  or 


<»l 


408  NA8MYTH    ON    TOOLS 

screw  on  the  work ;  and  according  to  the  respective  disp 
meters  of  the  wheels  w  w,  so  shall  we  have  a  screw  formed 
on  X,  more  or  less  fine  m  the  pitch  of  the  thread,  accord- 
ing to  the  proportions  of  the  respective  diameters  of  the 
wheels  ww,  as  in  the  figure  w  or  the  work,  is  twice  the 
diameter  of  w  or  the  end  of  the  slide  screw.  The  pitch 
of  the  thread  on  x  will  he  twice  as  wide  as  on  s,  and  as  8 
and  X  are  revolving  in  opposite  directions,  we  shall  have  a 
right  hand  screw  on  the  one,  and  a  left  hand  screw  on  the 
other,  or  the  reverse,  according  to  the  nature  of  the  guide 
screw  s ;  and  by  placing  an  intermediate  wheel  between  w 
and  w,  we  shall  then  cause  them  to  be  either  both  right 
hand  screws,  or  both  left,  as  the  case  may  be ;  the  depth 
of  cut  is  given  in  succession,  by  the  set  or  transverse  ad- 
justing screw  N. 

502.  Again,  in  the  case  of  the  wheel-cutting  machine, 
we  have  the  slide  rest  in  full  existence.     See  Fig.  6. 

503.  All  wheel-cutting  machines,  however  complex  they 
may  be  in  their  minor  arrangements,  consist  of  two  essen^ 
tial  parts,  the  slide  rest  s,  which  holds  the  revolving  cutter 
R,  and  the  spindle  t,  on  which  the  wheel  w,  which  has  to 
be  cut,  is  fixed.  This  spindle  is  made  part  of  the  dividing 
wheel  D,  by  fitting  into  a  socket  or  chuck,  so  that  when 
the  head  d  is  moved  roimd  in  successive  steps,  or  according 
to  the  required  divisions  on  the  face  of  it,  which  is  set  off 
or  divided  and  held  fast  by  the  index  point  or  holder  e,  it 
is  evident  that  whatever  be  the  di>dsions  or  fractions  of  the , 
divided  circle  d,  we  move  round  step  by  step ;  the  same 
will  be  most  faithfully  transferred  to  the  wheel  w,  which 
we  desire  to  cut  or  divide  into  teeth ;  and  by  means  of  die 
slide  rest  s,  we  slide  the  revolving  cutter  across  the  bee  or 
edge  of  the  wheel  w.  It  is  likewise  evident,  that  we  must 
thereby  cut  a  tooth  every  time  we  slide  the  cutter  across, 
after  each  division  is  taken  in  succession  by  the  ahifting  gf 
the  head  or  dividing  wheel  d. 


AND    MACHINES. 


This  is  a  very  meagre  descripdon  of  the  principle  of  a 
moBt  important  machine,  in  which,  as  in  innumerable  other 
iiHtances,  the  slide  principle  enables  us  to  produce  with  such 


410  NASMYTH    ON    TOOLS 

facility,  results  in  the  form  of  workmanship,  whose  mathe- 
matical accuracy  throws  all  hand  work  utterly  into  the  shades 
not  only  as  to  absolute  precision,  but  also  economy  of  pro- 
duction. 

504.  As  before  said,  were  I  to  endeavour  to  trace  in 
detail  the  countless  applications  of  the  slide  principle  firom 
its  first  appearance  before  the  mechanical  world,  as  intro- 
duced by  the  late  celebrated  Henry  Maudslay,  and  follow 
it  down  to  the  present  time,  a  thousand  pages  would  not 
give  space  for  all  that  might,  with  such  truth  and  justice, 
be  said  on  the  advantages  which  mankind  have  been  and 
are  now  deriving  from  the  slide  rest,  and  its  lineal  de- 
scendants. 


505.  Some  Observations  respecting  the  Form  of  Tools 
employed  for  Turning  and  Planing  Irony  Brass, 
^c.y  toget/ier  with  some  Remarks  on  the  Hardening 
and  Tempering  of  sicch  Tools. 

Hitherto,  so  far  as  I  am  aware,  the  form  of  tools  em- 
ployed in  turning  or  planing  iron,  &c.,  has  not  either  re- 
ceived that  attention  which  the  importance  of  the  subject 
calls  for,  nor  has  any  attempt  been  made  to  reduce  the  sub- 
ject to  such  plain  and  general  principles  of  which  it  is  not 
only  capable,  but  when  so  treated,  then  only  adapted  to  be 
of  service  to  those  in  whose  hands  the  management  of  such 
tools  is  for  the  most  part  entrusted.  Indeed,  so  much 
practical  importance  attaches  to  this  subject,  that  the 
quality  as  well  as  the  quantity  of  work  produceable  firom 
turning  lathes  and  planing  machines,  entirely  depends 
upon  the  skill  of  the  operator  in  giving  to  his  tools  the 
proper  form.  There  are  many  excellent  workmen,  wh0|  by 
a  species  of  intuition,  have  acquired  the  art  of  giving  to 
the  tools  either  the  true  form,  or  so  near  have  they  got 


AND    MACHINES.  411 

true  principlp,  that  by  holding  to  and  repeating 
again  and  again  that  tbrm  which  they  found  the  best,  they 
are  enabled  to  produce  the  required  resuh.  But  even 
with  such,  when  a  case  occurs  in  which  they  have  to  go  a 
little  out  of  their  usual  routine,  they  are  then  as  much  "  at 
sea"  as  if  they  knew  nothing  about  the  matter.  This 
ftrises  from  no  other  cause  than  the  want  of  the  knowledge 
of  the  general  principle,  which  would  guide  them  to  the 
true  form,  whatever  be  the  case  ;  and  moreover,  now  that 
slide  lathes  and  planing  machines  are  becoming  so  very 
common  in  the  workshops  of  engineering  establishments, 
and  that  such  machines,  from  their  automaton  power,  no 
longer  require  regularly  bred  mechanics  to  attend  them,  it 
becomes  more  than  ever  necessary  to  reduce  the  subject  to 
those  simple  principles  to  which  it  is  capable,  so  that 
ithe  subject  may  be  brought  within  the  range  of  the  sup- 
•^sed  inferior  capacity  of  a  humbler  grade  of  men,  from 
whom  we  want  no  more  than  careful  attention  to  secure  the 
Iwst  results  from  those  surprisingly  productive  machines. 
Wo  shall  now  proceed  to  the  subject  of  these  remarks,  and 
with  that  view  shall  take,  in  the  first  place,  the  most  sim- 
ple case. 

The  chief,  and  indeed  the  only  point  which  we  require  to 
consider,  is  the  direction  in  which  we  wish  to  cut  or  pene- 
trate the  metal.  Suppose,  therefore,  the  plane  a  b  is  the 
surface  of  a  plane  of  metal,  from  which  we  wish  to  cut  off 

Iavings,  in  the  direction  a  b,  either  hy  a  u  moving  against 


I 
I 

I 


% 


li 


\g) 


41 S  NASMTTH    ON   TOOLS 

the  tool,  or  the  reverse,  namely,  the  tool  moving  against  it^ 
for  it  is  the  same  action  in  either  case.  Suppose  we  were  to 
employ  such  a  tool  as  No.  1 ;  in  this  case  we  should  have 
little  or  no  penetrating  quality  in  the  form  of  the  tool, 
which  would  in  consequence  not  cut,  but  rvh  off  the  par- 
ticles, or  crush  them  off  by  sheer  brute  force.  The  reason 
of  this  is,  that  we  have  given  it  so  very  blunt  or  obtuse  an 
edge  at  the  point  of  cutting,  that  by  their  coming  against 
it  at  right  angles  to  its  face,  the  whole  force  which  moves 
the  plane  a  b  will  be  consumed  in  merely  rubbing  off  (not 
cutting)  the  particles  of  metal. 

Next,  in  the  case  of  No.  2,  which  looks  more  like  a  tool 
that  would  cut,  we  shall  find  that  there  again  we  should 
fail  to  produce  the  required  result,  and  also  encounter  other 
evils.  In  this  case  we  still  have  no  more  penetrating  pro- 
perty in  the  direction  a  b,  for  the  force  of  the  tool  is  still 
m  the  same  position,  with  regard  to  the  surface  to  be  cat» 
as  in  the  instance  of  No.  1,  that  is,  it  is  at  right  angles  to  it^ 
so  that  we  have  no  advantage  here ;  and  what  is  far  worsen 
we  have  from  this  tool  a  penetrating  quality,  in  a  direction 
quite  opposite  to  that  which  we  desire,  namely,  in  the  di- 
rection c  D.  In  moving  the  surface  a  b  against  this  tool. 
No.  %  we  should,  on  attempting  to  take  a  cut,  find  that 
the  penetrating  quality  in  the  direction  c  d,  would  imme- 
diately exhibit  itself  in  a  series  of  saw,  teeth-like  marks, 
more  or  less  deep,  according  to  the  strength  of  the  cut  and 
that  of  the  tool,  which  indeed  would,  on  account  of  its  form, 
not  preserve  its  point  entire  for  a  moment,  but  would  be 
snipped  off  with  little  or  no  force,  because  the  cross  section 
of  metal  at  its  point  is  scarce  measurable.  This  is  the 
most  usual  error  in  the  forming  of  tools,  that  is  to  say,  be- 
cause they  look  sharp,  that  is  thought  sufficient ;  forgetting 
altogether  the  direction  in  which  the  strain  is  to  be  ap- 
plied, and  in  consequence  not  providing  sufficient  metal 
a  cross  section  in  the  direction  of  the  strain. 


.^ND    SIU'HINES. 


413 


^ 


If  we  look  to  No.  3,  we  shall  find  that  all  these  requi- 
jritcs  are  provided.  In  the  first  place  we  have  a  high  de- 
gree of  a^-uteness  in  the  direction  of  the  cut,  namely  ab  ; 
then  as  to  strength,  behind  the  point  we  have  all  the  nietaJ 
from  E  to  F  to  give  the  point  E  the  requisite  support ;  in 
short,  as  regards  strength,  we  have  as  much  more  strength 
in  the  case  of  No.  3,  over  No.  %  as  the  distance  e  f  is 
greater  than  c.  No.  2.  Besides  this  great  strength  which 
we  have  in  the  case  of  No.  3,  we  have  also  another  advan- 
tage of  great  moment,  namely,  the  entire  absence  of  all 
tendency  to  chatter  or  produce  a  rippled  surfece,  fe  acting 
as  a  most  complete  stop  to  ajiy  risk  of  digging  into  the  sur- 
fcce  which  we  are  planing  or  turning,  which  would  in- 
evitably be  the  case  with  No.  2,  supposing  the  point  to  be 

tpable  of  resisting  the  force,  which  it  could  not.  The  very 
of  the  shaving  in  the  case  of  either  of  these  tools, 
'flTOald  exhibit  the  relative  advantages  of  each.    In  the  case 

No.  3,  they  would  be  most  complete  curls,  as  may  be 

dent  from  the  form  of  the  tool. 

In  No.  1,  therefore,  we  have  strength,  hut  no  acutcness 

either  direction, 

In  No.  2  we  have  acuteness,  it  is  true,  but  in  a  direction 
quite  opposite  to  that  in  which  we  require  it,  and  7to 
strength. 

In  No.  3  we  have  acuteness  entirely  in  the  direction 
in  which  we  require  it,  and  the  greatest  degree  of 
strength. 

Wo  may  therefore  establish  from  this  attempt  at  inves- 
ligation,  the  following  principle,  namely,  that  in  forming 
and  setting  a  tool  to  cut  any  surface,  we  have  only  to  attend 
4o  placing  it  so  that  the  end  of  it  forms  the  least  possible 
angla  with  the  surfiire  to  be  cut,  and  wlxatever  degree  of 
acuteness  be  considered  proper,  let  the  keenness  be  given 

hollowing  out  the  surface  e  c,  as  given  here. 


414 


NASMYTH    ON    TOOLS 


No.  3. 


I  again  repeat  the  principle,  namely,  that  in  forming  the 
cutting  tool,  what  we  have  to  attend  to  is,  to  let  the  end  of 
the  tool  he  as  nearly  parallel  to  the  surface  to  he  cut  as 
possible,  and  any  acuteness  that  may  be  required  shall  be 
given  to  the  surface  on  which  the  shavings  slide ;  the  very 
same  holds  good  in  the  case  of  turning  tools,  and  indeed 
in  every  tool,  from  a  razor  or  carpenter's  chisel  up  to  the 
most  enormous  and  powerful  tool  in  a  lathe  or  planing  ma- 
chine. In  the  case  of  turning,  we  may  sec  the  application 
of  the  "  principle  '*  very  clearly  exemplified. 


No.  «3  as  a  turning  tool. 


Here  we  see  No.  3  as  a  turning  tool,  ab  being  a  portion 
of  a  cylindrical  bar  in  the  lathe ;  £  f  should  be  as  near  as 
possible  a  tangent,  that  is,  at  right  angles  to  the  radius  of 
the  curve. 


AND    MACHINES. 


415 


No.  2. 


In  the  case  of  No.  %  employed  as  a  turning  tool,  we 
should  not  be  able  to  preserve  its  point  for  an  instant,  as 
will  be  evident  from  the  small  cross  section  at  c. 

When  scraping  is  all  that  is  necessary,  which  is  a  last 
finish  just  before  preparing  the  work  to  be  polished,  No.  1 
may  be  employed  with  advantage,  as  in  that  case  its  low 
penetrating  quality  in  both  directions  becomes  of  much  ser- 
vice, but  then  it  is  not  desired  to  employ  it  as  a  cutting 
tool. 


s^i^-^^^^;, 


■K^N^;^§:^#. 


In  the  instance  of  a  common  joiner's  plane,  we  shall  find 
the  same  principle  carried  out  most  fully ;  e  f  is  the  plane 
iron,  A  B  being  as  before  the  surface  to  be  cut.  In  the  case 
of  this  tool,  an  artificial  end  is  given  to  the  cutting  tool, 
by  means  of  the  sole  of  the  plane,  which  gives  the  requisite 
non-penetrating  quality  in  all  directioiis,  except  Uiat  in 


416 


NASMYTH    ON    TOOLS 


which  we  require  to  remove  the  material,  or  take  the  cat, 
namely,  ab. 


A 


The  same  again  is  seen  in  the  action  of  a  chisel  or  hat- 
chet. It  will  be  observed  that  the  bevelled  surface  of  the 
chisel  is  always  placed  outwards,  and  the  flat  surface  placed 
next  to  the  wood  which  we  are  about  to  cut,  so  that  the 
angle  between  the  face  of  the  chisel  next  the  wood,  and 
the  sur^c  of  the  wood,  shall  form  the  least  possible  an^ 
with  it. 


AUo,  in  forming  drills,  we  shall  find  the  very  same  prin- 
ciple in  action,  as  has  been  given  in  the  foregoing  ex- 
amples. Thus,  H  being  the  end  view  of  a  drill,  the  e^gv 
UP  should  be  the  least  possible  prominent,  or  oat  of  t^ 


AND    MACHINES. 


417 


plane  of  the  sutfiace  of  which  they  are  the  edges,  o  s  heing 
less  prominent  than  o  p,  so  that  there  may  he  as  little  pe- 
netrating quality  at  the  edge  op  as  possible.  A  drill  so 
formed,  will  cut  the  smoothest  holes  without  any  chatter- 
ing, which  is  so  commonly  the  case  when  the  edges  are 
bevelled  very  much  back,  as  given  at  r.  Such  a  drill  would 
very  soon  lose  its  edge,  and  would  cut  a  very  rough  hole 
besides. 

In  order  to  give  great  keenness  to  the  edge  of  the  drill, 
we  have  only  to  apply  the  same  principle  as  before  stated, 
in  respect  to  turning  tools,  by  hollowing  out  a  groove  at  x, 
on  each  cutting  face. 

Pig.  1. 


Face  Tool  applied  to  the  Gauge, 


FlO.    2. 


K 

Right  hand  To(»l 

ZX        i 

Face  Tool,                                       | 

1                                                                                                                       /                         • 

Lf/t  hand  Tool,                                              /                          \ 

V     1 

The  above  is  a  sketch  of  a  very  convenient  and  simple  tool 
gauge,  for  enabling  any  one  to  ascertain  whether  a  tool  is 
ground  or  formed  to  the  proper  angle.  It  consists  of  a  planed 
plate  of  metal,  ab,  on  whose  surface  there  is  at  one  end  fixed 
a  conical  steel  pin  c,  whose  taper  or  angle  formed  by  the 
sides  of  the  cone  with  the  surface  of  the  plate  ab,  is  just 
that  which  is  proper  for  the  cutting  face  of  the  tool  c,  be- 
ing  a  cone  given  in  a  very  simple  universal  gauge  for  every 


418  NASMYTH    ON    TOOLS    AND    MACHINES. 

kind  of  tool,  such  as  seen  in  Fig.  S.  By  using  this  ganger 
all  difficulty  of  forming  the  tools  to  the  proper  angle,  is  at 
once  removed. 

And  the  same  gauge  will  answer  for  every  kind  of 
planing  or  turning  tool  whatsoever,  and  of  whatever  size. 
A  B  may  he  ahout  15  inches  long,  hy  5  wide,  and  about  |th8 
of  an  inch  thick ;  these  dimensions  are  by  no  means  abso- 
lutely requisite,  but  will  be  found  generally  usefiiL 

The  angle  formed  by  the  sides  of  the  cone,  and  the  sur- 
face of  the  plate,  should  be  about  three  degrees. 


GENERAL 


EXPLANATION   OF  THE   PLATES 


PLATE  I.— ESSAY  I. 


On  thb  Teeth  of  Wheels. 


Chart  shewing  the  horses'  power  to  which  the  teeth  of  wheels  of  certain 
pitches,  working  under  different  circumstances,  are  equal, — is  described 
on  the  Plate,  and  in  Chap.  V.  Arts.  163—179. 


PLATE  IL— ESSAY  II. 
On  the  Shafts  of  Mills  and  other  Machines. 

Fig.  1. — No.  1,  No.  2,  and  No.  3. — Old  mode  of  fixing  gudgeons  called 

laid-in-pudgeons^  Art.  187,  pp.  178,  179. 
Pig.  2. — No.  1  and  No.  2. — Cross-tailed  gudgeons.  Art.  188,  p.  179. 
Fig.  3,  Fig.  4,  and  Fig.  5. — Parts  of  cross-tailed  gudgeons,  Art.  188, 

p.  179. 
Fig.  6. — A  hollow  cast  iron  shaft.     No.  2.  An  end  view,  to  shew  the 

manner  of  fixing  the  gudgeons.  Art  192,  p.  181. 


PLATE  III.— ESSAY  II. 


Fig.  7. — No.  1,  No.  2,  No.  3,  and  No.  4. — ^A  feathered  shaft,  with  dif- 
fefooi  seetjonsy  Art  193,  p.  181. 


T.'-7 


4520  EXPLANATION    OF   THE    PLATES. 

Fig.  8. — No.  1  and  No.  2. — Square  shafts,  Art.  193,  p.  182. 

Fig.  9. — No.  1  and  No.  2. — Stress  on  shafts.  Art  194,  pp.  182,  183« 

Fig.  9. — Water  wheel  with  teeth  on  the  shrouding,  to  give  motion  to  Ae 

mill,  Art.  104,  p.  183. 
Fig.  10. — The  cliief  strain  on  the  vertical  shaft  is  that  which  tends  to  twiit 

it.  Art.  194,  p.  183. 
Fig.  11. — Example  of  a  compound  strain  on  the  shaft,  Art.  194,  p.  183. 
Fig.  12.— Shaft  of  a  water  wheel.  Art.  194,  p.  183,  and  Art  195,  p.  188. 

PLATE  IV.— ESSAY  II. 

Fig.  1 . — Effect  of  position  in  causing  a  greater  or  less  degree  of  stnin  on 

shafts.  Art.  195,  p.  183. 
Fig.  2. — Stress  on  shafts  from  the  action  of  the  moving  power.  Art  196, 

p.  184. 
Fig.  3,  Fig.  4.  No.  1,  No.  2,  No.  3;   and  Fig.  5.— Effect  of  position  in 

causing  more  or  less  stress  on  the  sliafts.  Art.  197,  pp.  184,  185;  and 

Art.  198,  pp.  18.5,  186. 
Fig.  6  and  Fig.  7. — To  explain  the  effect  of  the  place  of  a  wheel  on  a 

shaft.  Art.  199,  p.  186. 

PLATE  IV.  A.— ESSAY  II. 

Fig.  1,  Fi«^.  2,  Fig.  .'3. — Represent  an  improved  method  of  fixing  gndgeons 
on  wooden  shafts  Art.  207,  p.  1J^3,  &c. 

Fig.  4. — Plan  of  a  sliaft  and  wheels  to  show  how  the  stress  on  a  shaft  may 
bo  determined,  Art.  207,  pp.  1 93,  1 94. 

Fig.  .*>,  Fig.  fi,  and  Fig.  7. — Figures  to  show  the  stress  on  shafts  and  gud- 
geons in  different  circumstances.  Art.  210,  p.  19,3. 

PLATE  v.— ESSAY  III. 

On  the  Longitudinal  Connexion  of  Shafts,  denominated 

Couplings. 

Fig.  1. — Represents  three  shafts  connected  hy  couplings,  supported   by 

double  bearings.  Art.  291,  p.  2GG. 
Fig.  2. — No.  1  and  No.  2. — The  square  coupling,  witli  double  bearings^ 

Art.  202,  p.  207. 
No.  3. — A  different  modification  of  the  square  coupling,  Art.  292, 

p.  267. 
No.  4. — A  coupling  box  made  in  two  pieces,  Art  298,  p.  29T* 


EXPLANATION  OF  THE  PLATES.  421 

Fig.  3. — No.  1,  No.  2,  and  No.  3.— A  variety  of  the  couplings  with 

doable  bearings,  Art.  294,  p.  268. 
Fig.  4. — No.  1,  No.  2,  and  No.   3. — The  round  coupling,   Art.  295, 

pp.  268,  269. 
Fig.  5. — No.  1  and  No.  2. — Clutches  or  glands.  Art  297,  p.  269. 
Fig.  6. — No.  1  and  No.  2. — Boring-mill  clutch,  first  construction.  Art. 

300,  pp.  270,  271. 
Fig.  7. — No.  1  and  No.  2. — Boring-mill  clutch,  second  construction,  Art. 

303,  pp.  271,  272. 
Fig.  8. — No.  1,  No.  2,  and  No.  3. — Coupling  having  circular  plates,  Art 

306,  p.  272. 

PLATE  VI.— ESSAY  III. 

Fig.  9. — No.  1  and  No.  2. — Coupling  link  used  by  Messrs.  Boulton  and 
Watt  in  their  portable  steam  engines,  Art  308,  p.  273. 

Fig.  1 0. — No.  1 ,  No.  2,  and  No.  3. — A  coupling  sometimes  used  to  con- 
nect the  fly-wheel  shaft  of  a  steam  engine  with  the  mill-work,  and  is  so 
contrived,  that  in  case  the  fly  should  turn  in  a  wrong  direction,  the  mill- 
work  remains  at  rest,  Art  310,  p.  273. 

Fig.  11. — The  universal  joint.  Art.  314,  p.  276. 

Fig.  12. — No.  1,  No.  2,  and  No.  3. — Square  coupling,  having  one  bearing. 
Art.  318,  p.  277. 

Fig.  13. — No.  1  and  No.  2. — Mode  of  coupling  rollers  used  in  machinery 
for  spinning  cotton.  Art.  320,  p.  278. 

Fig.  13*. — No.  1,  No.  2,  and  No.  3. — Cylindrical  coupling,  having  one 
bearing.  Art.  321,  p.  279. 

Fig.  14. — Bolted  coupling,  used  in  some  mills  in  Manchester,  Art.  323, 
p.  279. 

PLATE  VII.— ESSAY  IIL 

Fig.  15. — No.  1,  No.  2,  and  No.  3. — A  variety  of  the  bolted  coupling, 
Art.  325,  p.  280. 

Fig.  16.-^Another  modification  of  the  bolted  coupling.  Art  326,  p.  280. 

Fig.  17. — No.  1,  ^o.  2,  and  No.  3. — Quadrant  coupling,  Art  327,  p.  280. 

Fig.  18. — No.  1,  No.  2,  No.  3,  and  No.  4. — Coupling  forming  an  uni- 
versal joint,  Art.  329,  p.  281. 

Fig.  19. — No.  1. — Elevation  of  coupling,  executed  at  Manchester,  Art 
331,  p.  282.     No.  2,  and  No.  3,  are  on  the  next  Plate. 


422  EXPLANATION    OF   THE   PLATES. 


PLATE  VIIL— ESSAY  III. 

Fig.  19. — No.  2  and  No.  3. — Sections  of  couplings  czeeated  at  Mm- 
Chester,  Art.  331,  p.  282.     No.  1,  is  on  tibc  preceding  Plate. 

Fig.  20. — No.  1,  No.  2,  and  No.  3. — Cylindrical  coupling,  having  projec- 
tions, Arts.  334,  335,  pp.  282,  283. 

Fig.  21. — Square  coupling  (having  a  coupling-box)  for  upright  ahafta,  Alt. 
337,  p.  283. 

Fig.  22. — No.  1  and  No.  2. — Square  coupling  for  upright  ahafta,  honng  a 
socket  instead  of  a  box,  Art.  338,  p.  284. 

Fig.  23. — Coupling  for  upright  shafts,  having  projecting  quadranta,  Alt. 
340,  p.  284. 

Fig.  21'. — Lying  shafts,  shewing  the  situation  of  the  couplings  with  regpid 
to  the  wheels.  Art.  348,  p.  286. 


PLATE  IX.— ESSAY  IV. 

On  Methods  of  Disenoaoing  and  Re-enoaoino  Machinsbt,  while 

IN  Motion. 
Fig.  1,  and  No.  2. — The  sliding  pulley.  Art.  362,  p.  294.      « 
Fig.   2.— No.  1,  No.  2,  No.  3,  and  No.  4.— The  bayonet.  Art.  894, 

p.  295. 
Fig.  3.— No.  1  and  No.  2.--The  lock  pulley,  Art.  3G6,  p.  297. 


PLATE  X.— ESSAY  IV. 

Fig.  4. — No.  1  and  No.  2. — Fast  and  loose  pulleys,  or  dead  and  live  pul- 
leys, Art.  3G8,  p.  297. 

Fig.  5.— Sack  tackle,  Art.  372,  p.  299. 

Fig.  6. — No.  1. — Wheels,  in  gear^  having  a  moveable  bridge.  Art.  371,  p. 
298.     No.  2,  is  on  the  next  Plate. 


PLATE  XL— ESSAY  IV. 

Fig.  6. — No.  2. — Wheels  out  of  gear ^  Art.  374,  p.  299.     No.  1,  ia  on  the 

preceding  Plate. 
Fig.  7. — No.  1  and  No.  2. — Clutch  for  engaging  the  wheel  ▲  with  the  abaft 

B,  Art.  376,  p.  300. 
Fig.  8.— No.  1  and  No.  2.— Friction  clutch,  Art  378,  p.  301.     No.  3  ia 

on  the  next  Plate. 


EXPLANATION    OF   THE    PLATES.  423 

PLATE  XII.— ESSAY  IV. 

Fig.  8. — No.  3. — Hoops  of  friction  clutch,  Art.  378,  p.  302.     No.  1  and 

No.  2  are  on  the  preceding  Plate. 
Fig.  9. — No.  1  and  No.  2. — Friction  cones,  Art.  380,  p.  302. 
Fig.  10. — No.  1  and  No.  2. — Wheels  moying  hy  contaction.  Art.  382,  p. 

303. 
Fig.  11.— Sack  tackle.  Art.  384,  p.  304. 
Fig.  12. — No.  1   and  No.  2. — Self-disengaging  coupling.  Art.  386,  pp. 

304,  305. 

PLATE  XIII.— ESSAY  V. 

On  Mbchanism  fob  Equalizing  thb  Motion  of  Mills,  denominated 
Lift-Tbntebs,  Engine  Govebnobs,  and  Watbb- Wheel  Govebnobs. 

Fig.  1. — Steam-engine  governor.  Art  389,  p.  308. 

Fig.  2. — The  balls  a  b  and  c  being  made  to  revolve,  are  all  found  on  the 

same  horizontal  plane.  Art  392,  p.  310. 
Fig.  3. — LifWtenters  for  wind  mill,  first  construction.  Art  396,  p.  311. 
Fig.  4.— -Lifl-tenter,  second  construction.  Art.  397,  p.  312. 
Fig.  5. — No.  1.-— Elevation  of  water-wheel  governor,  first  construction, 

Art.  399,  p.  313.     No.  2,  No.  3,  and  No.  4,  of  Fig.  5,  are  on  the  next 

Plate. 

PLATE  XIV.— ESSAY  V. 

Fig.  5.— No.  2,  No.  3,  and  No.  4. — Water-wheel  governor,  first  construc- 
tion. Art.  399,  p.  314.     No.  1,  of  Fig.  5,  is  on  Plate  XIII. 

Fig.  6. — No.  1  and  No.  2. — ^Water-wheel  governor,  second  construction. 
Art.  401,  p.  315. 

PLATE  XV.-ESSAY  V. 

Fig.  7. — Water-wheel  governor,  third  construction.  Art.  402,  p.  315. 

Fig.  8.— No.  1  and  No.  2. — ^Wator-wheel  governor,  fourth  construction, 
Art  403,  p.  316. 

Fig.  9. — No.  1,  No.  2,  and  No.  3. — Water-wheel  governor,  fifUi  construc- 
tion. Art.  404,  p.  317. 

PLATE  XVI.— ESSAY  VL 

On  Changing  the  Velocity  of  Machineby  while  in  Motion. 

Fig.  1. — ^Turning  lathe  motion.  Art  419,  p.  335. 
Fig,  2.-* Alternate  cones,  Art  421,  p.  336. 


4S4  EXPLANATION  OF  THE  PLATES. 

Fig.  3. — Wheels  moving  by  contaction,  Art.  423,  p.  336* 

Fig.  0. — No.  1  and  No.  2. — Mechanism  used  in  cotton  spinning  odled 

double  speed,  third  construction,  Art  480,  p.  340. 
Fig.  4  and  Fig.  5,  are  on  Plate  XVII.— Art  426,  p.  338. 


PLATE  XVII.— ESSAY  VL 

Fig.  4. — Double  speed,  first  construction.  Art  426,  pp.  338,  339. 
Fig.  5.— Double  speed,  second  construction.  Art  426,  pp.  338,  339* 


PLATE  XVIIL-KSSAY  VII. 

On  tbb  Fraiiino  of  Mill-Wokk. 

Fig.  1. — Pivot  of  upright  shafts,  Art.  439,  p.  346. 

Fig.  2. — No.  1,  No.  2,  and  No.  3.— Pivot  of  upright  shaft  Yamng  a  sted 

foot.  Art  440,  p.  346. 
Fig.  3. — Cylindrical  steel  pivot.  Art.  441,  p.  347- 
Fig.  4. — No.  1,  No.  2,  and  No.  3. — Journal  of  tlpright  shaft  supported  by 

a  breast,  Art  444,  p.  347. 
Fig.  5. — Millstone  bush.  Art.  445,  p.  347. 

Fig.  6. — No.  1,  No.  2,  and  No.  3. — Headstock  framing  for  supporUng  gud- 
geons of  water  wheel,  Art.  446,  p.  348. 
Fig.  7. — No.  1,  No.  2,  No.  3,  and  No.  4. — ^Wooden  framing  for  lying 

shafts,  Art.  447,  p.  348. 
Fig.  8. — Framed  post.  Art.  448,  p.  349. 
Fig.  9. — Framing  for  lying  shafts  suspended  from  a  ceiling,  Art  448,  p. 

349. 
Fig.  10. — No.  1  and  No.  2. — Bridge  and  pedestal  of  upright  shaft,  Art 

449,  p.  349. 
Fig.  11. — Wooden  framing  of  an  upright  and  lying  shaft  connected  by  borel 

wheels.  Art  450,  p.  349. 
Fig.  12. — Framing  having  cloves,  Art.  451,  p.  349. 
Fig.  13  and  Fig.  14. — Transverse  sections,  showing  forms  of  pieces  of  timber 

and  of  iron.  Art.  456,  p.  351. 
Fig.  15. — Feathering,  Art.  457,  p.  352. 
Fig.  16,  Fig.  17,  and  Fig.  18. — Represent  advantageous  forms  of  sections, 

Art.  457,  p.  352. 
Fig.   19. — No.  1  and  No.  2. — Bridge  made  of  wood  for  sustaining  lying 

shaft.  Art  458,  p.  353. 
Fig.  20. — No.  1  and  No.  2.— Bridge  made  of  east  iron  for  nwtsining  lying 

shaft,  Art  458,  p.  353. 


EXPLANATION    OF   THE   PLATES.  425 

PLATE  XIX.-ESSAY  VII. 

Fig.  21 Headstock  for  water  wheel,  Art.  460»  p.  353. 

Fig.  22. — Mode  of  snspending  lying  shafts  by  cast  iron,  Art.  461,  p.  353. 
Fig.  23. — No.  1  and  No.  2. — Cast  iron  framing  for  supporting  3  pair  of 

flour-mill  stones,  Art  462,  p.  353. 
Fig.  24. — No*  1  and  TSo.  2.—- Machine  palled  iqu$ezer$j  used  in  bleaching, 

haying  cast  iron  framing,  Art  462,  p.  353. 


EXPLANATION 


OF 


THE    ADDITIONAL    PLATES 

OF  THIS  EDITION, 
NUMBERED  AND  DESCRIBED  FROBI  PLATE  XIX. 


PLATE  XX. 


Represents  several  diagrams,  shewing  the  theoretical  mode  of  describing 
tlie  teeth  of  wheels ;  to  accompany  (the  Appendix  A.)  Professor  Willis s 
Paper  on  that  subject,  pp.  149—157,  and  166—172. 

PLATE  XXL 

Bbahah's  original  Slide  Rest. 

Fig.  1.  is  an  elevation;  Fig.  2.  an  end  view;  and  Fig.  3.  a  plan  of 
Bramah's  slide  tool,  which  was  first  used  in  the  year  1 794,  and  was  the 
workmanship  of  the  late  Mr.  Maudslay,  made  by  himself  when  in  the 
employment  of  the  late  Mr.  Bramah. 

The  work  to  be  turned  is  placed  in  the  usual  way  between  two  centxe 
pieces,  one  is  shewn  in  Figs.  1  and  3,  by  Cy  where  it  is  secured.  The  square 
bar  dy  on  one  end  of  which  is  fixed  the  tool,  is  then  made  to  advance  by 
means  of  the  screw  e,  working  through  a  nut,  secured  to  the  square  bar  d 
by  a  small  set  screw ;  this  nut  piece  can  be  moved  along  the  bar  according 
to  the  length  required,  and  by  turning  the  handle  e,  the  bar  <f,  and  conse^ 
quently  the  tool,  is  brought  close  to  the  work  which  it  is  intended  to  turn. 
The  frame  for  carrying  this  bar  slides  between  two  V's  on  the  frame  b,  by 
the  motion  given  to  the  handle  and  screw ;  thus  it  will  be  seen,  this  restbas 
two  motions,  the  one  at  right  angles  to  the  other. 

The  slide  rest  and  frame  h  are  moveable  along  the  traversing  bar  a, 
according  to  the  length  of  the  work,  and  when  placed  in  any  particular 
position  may  be  secured  by  the  handle  and  screw  underneath. 

This  rest  is  very  different  to  those  in  use  at  present,  as  will  be 
hereafter. 


EXPLANATION    OF    THE    PLATES. 


PLATES  XXII.  AND  XXIII. 

BuAMAU's  Laths  fdh  tubkino  Sphebes. 

The  Iidl  or  sphere  _;  is  fixed  in  the  usual  way  between  the  two  centres 
of  the  lathe,  having  projecting  pieces  on  it  for  that  purpose.  The  muidril 
is  driven  by  a  leather  strap  working  on  the  circumference  of  the  pulleys  or 
riggers  d;  oa  one  end  of  this  mandril  is  the  chuck  /,  having  a  pin  on  its 
face,  called  the  driver ;  when  this  begins  to  rerolve,  it  cornea  into  contact 
with  a  carrier  fixed  on  one  of  the  projecting  pieces  of  the  sphere,  which 
t  necessarily  drag  it  round  with  It,  and  thus  communicate  the  reTolving 
Lnotion ;  there  is  a  ECt  screw  on  this  carrier  to  fix  it  by,  as  is  shewn  in 
I  Figs.  1  and  3. 

The  tool  I,  Fig.  2,  is  fixed  to  the  moveable  slide  b,  by  two  adjusting 

ind  moves  on  its  centres  k'  h',  describing  the  circle  of  the  sphere ; 

L'tiib  slide  works  in  a  V  on  one  side  of  the  bed  a,  and  on  a  Rat  smooth  sur- 

I  Ace  on  the  other.     When  the  sphere  is  turned,  the  back  centre  screw  ^  is 

i  loosened,  which  at  once  liberatcx  it.     There  is  also  an  adjusting  screw  for 

the  bock  centre  of  the  mandril,  which  at  times  requires  tighleniug.     By 

withdrawiug  the  slide,  used  in  spherical  turning,  and  fastened  to  the  sliding 

carriage  by  four  screws,  a  common  band  rest  can  be  put  in  its  place,  when 

e  lathe  is  required  for  ordinary  purposes. 


PLATE   XXIV. 

Gbbat  Bokino  Lathe,  by  Messhs.  Nasmytr,  Gabkbll  and  Co. 

e  two  lathes  already  described  are  very  much  smaller  tlian  that  shewn 

y  this  Plate;  the  work  they  are  capuble  of  turning  is  altogether  of  a  different 

md. 

Figs,  1,  2,  and  3,  severally  represent  an  elevation  and  two  end  views, 

will  he  seen  that  this  lathe  is  suited  to  various  speeds,  depending  on 

B  dituneCera  of  the  piUleya  or  driving  riggers  rf,  which  are  put  in  motion  by  a 

'  stmp.     The  hcadstocks  a,  and  the  bed  A  ore  well  secured  to  the 

s  foundation,  by  means  of  strong  wrought  iron  holding  down  holts  for 

X  purpose ;  on  the  face-plate  c,  to  which  is  fixed  the  work  to  bo  turned, 

aa  internal  wheel  worked  by  a  small  pinion  hung  on  an  intermediate 

jbaft,  which  receives  its  motion  from  the  driring  shaft,  by  the  spur  wheel 

Bid  pinion  e  on  its  opposite  end;  long  mortices  are  shewn  on  the  face-plate 

D  Fig.  3,  through  which  the  work  to  be  turned  is  clamped,  and  revolves 

fith  it ;  /  is  the  slide  rest  for  carrying  the  tool,  placed  on  the  two  sUuidards 

J  tfKil  being  made  to  advance,  by  a  long  screw  the  length  of  the 

"■■  '111-  handle  seen  in  Fig.  1,  it  is  made  to  travel  idong  the  shding 


-.r* 


428         EXPLANATION  OF  THE  PLATES. 

bed.  It  has  also  a  second  motion  at  right  angles  to  tliat  already  described. 
The  tool  is  fixed  to  the  rest  by  four  screws,  Fig.  1.  The  headstock  a  it 
raised  on  a  carriage  b^  to  which  it  is  bolted.  This  hithe  is  of  large  dimeonMUy 
and  is  capable  of  boring  cylinders  of  great  diameters;  it  is  also  well  adapted 
for  turning  locomotive  engine  wheels. 

PLATE  XXV. 
Facb-turnino  Lathe,  by  Messrs.  Nasmtth,  Gabkbll,  and  Co. 

This  Plate  differs  but  little  from  the  last  described.  There  is  only  on 
rigger  or  pulley  a,  on  the  mandril  or  shaft,  and  the  face-plate  e  has  fbor 
adjusting  screws  for  securing  the  work  to  it  Instead  of  driving  this  plate 
from  the  pulley-shafti  it  is  put  in  motion  by  a  small  pulley  worked  hy  a 
strap,  from  the  main  shaft  of  the  building,  in  a  similar  way  to  the  pulley  d^ 
making  the  one  quite  independent  of  the  other.  The  internal  wheel,  and 
the  pinion,  are  similar  in  every  respect  to  those  described  by  Plate  XXIY., 
which  is  also  the  case  with  regard  to  the  slide,  the  slide-rest,  and  tbe 
headstocks  a  a. 

The  foundations  to  which  the  bed  b  is  bolted  are  of  stone  bedded  in 
concrete. 

PLATE  XXVL 

Foot-Lathe,  by  Mr.  F.  Lewis,  Manchester. 

The  bed  ^,  Figs.  1  and  2,  is  placed  on  the  two  standards  b  b,  forming  at 
the  same  time  centres  for  the  crauk-spindle  n  to  work  in,  which  receiTes 
its  motion  from  the  foot-board  m'  by  the  two  connecting  links  oa.     On  the 
spindle  n^  is  a  pulley  and  also  a  fly-wheel,  for  regulating  the  motion;  (the 
latter  being  used  at  times  as  a  pulley  for  quick  speeds;)  these  convey  it 
through  tlie  upper  pulleys  c,  by  a  leather  strap,  to  the  mandril  or  spindle  d^ 
on  the  end  of  which  is  a  chuck  or  face-plate.     The  headstock  ^  fonns  soit- 
ablc  bearings  for  tliis  spindle  to  revolve  in,  having  a  back  centre  eciew  to 
tighten  it  by.    The  following  headstock  r  slides  along  the  bed  to  any  position, 
according  to  the  length  of  the  work  to  be  turned,  which  is  placed  between  tbe 
two  centres  //;  tliat  in  the  latter  head:*tock  is  adjustable,  being  fixed  to  a 
cylinder  tf  working  in  it,  and  moved  either  backward  or  forward  by  the  handle 
or  wheel  (^\  which,  as  it  revolves,  works  a  screw  fixed  to  it  and  k^C  steady 
by  the  bolt  underneath.  Fig.  1 .    When  the  work  has  been  properly  eentevad, 
the  cvlinder  <;  is  made  secure  in  the  headstock  by  the  small  handle  and  acasw  A. 

The  slide  /  for  carrWug  the  tool  is  advanced  along  the  bed  hy  a  pimsa 
working  in  the  rack  i",  put  in  operation  by  a  small  handle  placed  osi  As 
square  end  of  the  spindle  •" :  a  motion  at  right  an^es  to  this  ia  oteiaad  ^ 


I 

JBC 

ft!' 

Lfotc 


EXPLANATION  OF  THE  PLATES.  420 

die  sliiley,  worked  in  the  game  way  by  the  ecrew  /,  and  lastly,  a  circulof 
motion  may  bo  given  to  the  loni  by  the  womi  and  worm-wheel  /.  The 
tool  is  placed  in  the  tool-box  «,  and  there  Rsed  by  the  two  set  eerews;  thia 
box  nmy  be  placed  in  any  position,  and  then  Becored  by  a  set-screw  m. 

An  index  y'  is  graduated  on  the  beadstock  g,  (at  the  purpose  of  tnrning 
spheres. 

The  rack  i'  is  secured  to  three  small  brackets  bolted  to  the  bed  of  the 
lathe: 

This  inaohino  is  capable  of  turning  a  bar  3  feet  long,  and  6  in.  diameter, 
and  spbericsl  cups,  halls,  &c.,  &c.,  5  in.  diameter,  having  an  index  to  turn 
either  conioally  or  longitudinally. 

As  will  be  seen  by  the  title,  this  lathe  is  put  in  motion  by  the  foot  of  the 

an  working  it,  and  unltke  tlic  last  two,  it  is  only  adapted  for  turning 
onall  work. 

PLATE  XXVII. 

Sbven-Feet  TuKNiNo-LAinB.  by  Mr.  F.  Lewis,  Manchesteh. 

In  the  usual  way,  the  top  part  of  the  fixed  bed  o.  Fig.  2,  is  of  the  V 
form,  upon  which  the  lieadstucks  bb  and  the  slide  h  are  moveable ;  that  for 
carrying  the  driving  apparatus  is  secured  to  it  by  strong  bolts  and  nuts ;  the 
■haft  of  the  driving  pulleys  c  works  in  the  two  bearings  of  the  headstock;  one 
d,  for  the  purpose  of  keeping  it  always  tight  and  to  avoid  any  play, 
rhich  would  be  the  case  were  it  parallel,  after  having  worked  any  length  of 
tightened  against  the  eone-piecc  of  the  beadstock  by  nuts  on  the 
ipodte  end  of  the  shaft.  To  the  face-plate  g,  die  work  is  fixed  by  four 
jaws  rrrry  moving  in  Vs  on  ihe  hack  of  the  plate,  and  radiating  to  the 
centre,  to  which  they  can  bo  approached  or  drawn  from,  by  a  small  handle 
the  square  end  of  the  screw,  by  which  means  they  are  made  to 
,  (lie  V'b  according  to  the  size  and  nature  of  the  work  to  he  turned ; 
motion  is  eommunicalod  to  the  face-plate  by  the  spur  pinions  and  wheels 
theintermediatcshofty,  and  thence  to  the  pinion  working  the  internal  w^hecl 
OB  tlie  lace-plute,  consisting  of  sis  segments  screwed  to  it.  By  this  arrange- 
ment the  Bjieed  of  t}io  face-plate  is  very  much  decreased,  and  it  slioutd  he 
understood  that  though  the  pUte  ff  is  on  the  main  sliaft,-  it  is  not  lixcd  to 
it  by  keys,  hut  runs  loose,  and  revolves  at  a  niucli  decreased  velocity ; 
B  triangular  carriage  is  bolted  to  the  headstock's  side  fur  supporting  the 
igB  of  the  intermediate  shaft. 

the  opposite  beadstock  the  oyltnder  q  and  centre  n  moves,  being  slid 
it  by  turning  the  wheel  and  screw  /,  according  to  any  required 
ition  for  centering  Uie  work,  trhich,  when  done,  is  firmly  secured  bv  the 
tightening  rings  and  nuts  y". 


430  EXPLANATION  OF  THE  PLATES. 

Tliis  headstocky  and  also  tlie  slide,  is  moved  along  the  sarface  of  tlie  bed 
a  hy  pinions  working  in  a  rack  o,  fixed  to  the  three  bmckots  p^  on  iti 
side.  Tlic  pinions  arc  made  to  revolve  hy  a  lever  turning  the  wheels  next  to 
them.  Independent  of  tlie  motion  of  the  slide  along  the  top  of  tho  bed, 
it  has  three  others,  viz. :  the  frame  j  can  swivel  round  on  the  carriage  t,  to 
any  angle ;  the  slide  X*  can  travel  along  the  frame  J  by  a  small  handle  fixed 
on  either  end  of  the  screw  k'y  and  lastly,  a  motion  at  right  angles  to  this 
is  obtained  by  the  slide  on  which  the  tool  is  placed,  being  also  worked  by  a 
screw ;  the  tool  is  fixed  by  the  bolts  l\  which  hold  it  firmly  in  its  plaoe« 

On  the  end  of  the  main  driving  shafl  e  is  the  centre  fi,  and  between 
this  and  the  other,  the  work  to  be  tiuncd  is  centered  in  the  usual  war. 

The  tires  for  locomotive  wheels,  and  also  wheels  of  a  diameter  of  seven 
feet,  can  be  turned  either  internally  or  externally  by  tliis  lathe. 

PLATE  XXVIII. 
Lathk  for  turning  Gun-Barrkls,  by  Messrs.  G.  and  J.  Rbnnib. 

The  object  of  this  machine  is  to  perform  an  operation  entirely  different 
from  that  ])rodiiced  by  the  several  motions  of  the  common  turning  lathe, 
where  the  work  is  either  of  a  cylindricid  or  elliptical  figure.  In  this  case, 
the  curve  required  to  be  turned,  on  the  whole  Icngtli  of  the  musket  barrel, 
is  of  an  irregular  form.  This,  as  will  bo  soeu  by  Figs.  1,  2,  3,  and  4  of 
this  Plato,  is  clfootod  hy  a  vory  simple  ooutrivuucc,  a  description  of  which 
fo]low.s : — 

Tho  l)od  //  is  secured  hy  tliroc  bolts  a'  u!  t\  to  the  side  frames  or  stand- 
ards a  a,  which  hnvo  also  fixed  to  thorn  a  tank  o  for  receiving  the  wngte 
oil  and  tho  iron  turnings  from  the  barrel  hoing  turned  ;  on  the  top  of  the  bed 
is  ])lacod  tho  hoadstock  fnuno  /,  for  carrying  the  several  jiarts  of  the  ma- 
oliinery ;  and  on  tho  mandril  if  arc  the  tight  and  loose  puller's  for  giving 
motion  to  the  machine :  the  centre  c'  of  this  mandril  is  adjusted  by  the 
hack  centre  c\  By  the  two  sets  of  spur  wheels  and  pinions  a  double 
motion  is  produced,  the  forward  hy  /  and  the  back>i'ard  by^,  which  are 
alternately  worked  by  means  of  the  small  clutch  ^,  thus  giving  a  revolving 
motion  to  the  square  threaded  screwy  for  advancing  the  tool  irame  /  along 
the  barrel  to  he  turned  ;  this  motion  is  made  self-acting  by  means  of  two 
small  bars  pp  fixed  to  the  clutch  lever,  having  projecting  pieces  on  their 
sides,  with  wliich  the  frame  /  comes  in  contact,  and  pressing  against  them, 
draws  them  and  engages  the  clutch  in  the  wheels  for  giving  the  oppoaitc 
motion,  while  the  object  of  the  handle  h'  is  to  disengage  it  by  hand.  The 
spindle  on  which  these  small  wheels  arc  fixed  runs  freely  in  brasa  bearings 
fitted  to  the  heodstock  f . 


I 


I 


on 


EXFLANATION    OF   THE    PLATES.  431 

On  the  opposite  end  of  the  machine  is  the  hcodstock  m,  haTing  bo  ftd- 
justnhle  centre  d,  worked  by  the  screw  and  handle  m",  and  may  be  fixed 
in  orv  required  position  by  a  small  set  screw  ?n"';  this  bead  can  be  moved 
kloiig  die  bed  b,  to  suit  the  length  of  any  barrel  n,  and  by  the  cross  bar  and 
traits  m'  may  be  secured  to  it.  The  musket  barrel  to  be  turned  is  slid  on 
the  long  bar  or  mandril  n',  which  it  fits  very  tightly.  One  end  of  tliis 
maiHlril  is  then  dropped  in  the  chuck  e  for  carrying  it  ronnd,  while  it  is 
centred  at  both  ends  in  the  centre  pieces  f'e'. 

The  most  curious  part  of  this  mnclunc  is  the  frame  I  for  holding  the 
tools  and  likewise  the  barrel  of  the  musket  being  turned ;  which  latter 
operadon  is  rendered  rather  difficult  from  the  irregular  shape  of  tlie  barrel. 
Fig.  3  shews  a  section,  looking  at  the  fmme  I,  which  consists  of  a  bock 
plate,  on  the  face  of  which  is  a  broAs  plate  with  the  eccentric  grooves 
shewn  by  tlie  dotted  lines  struck  from  different  centres ;  between  the 
latter  plate  and  the  back  plate  are  four  dies  sliding  in  V's,  having  ribs 
irorking  in  the  above  grooves,  which,  not  being  concentric,  press  them 
against  the  barrel,  and  thereby  hold  it  firmly  while  the  tools  V  are  turning 
it;  but  as  the  diameter  of  the  barrel  alters,  it  is  necessary  to  loosen  tlie 
dies,  in  order  to  allow  the  frame  /,  of  wliich  they  form  part,  to  slide  along 
the  bed  on  its  V's.  This  is  thus  effected :  a  curve  bar  or  template  k,  suit- 
ftblo  to  the  curve  required  by  the  barrel,  is  firmly  screwed  to  the  side  of  the 
bed,  its  npper  edge  i,  Fig.  3  and  4,  being  of  the  shape  of  a  V,  on  which 
the  lower  part  of  the  rack  f"  slides,  the  upper  part  working  at  the  same 
lime  in  the  V's  screwed  to  the  projection  given  to  the  back  plate  /  by  three 
•crews.  Thus  the  motion  produced  by  the  curve  of  the  bar  it  is  transmit- 
led  throngh  tlie  rack  and  the  segment  V  to  the  grooved  plate  on  which 
the  latter  is  fixed;  the  result  of  this  is  the  tightening  or  looseoing  the  four 
dies,  which  must  naturally  be  the  case,  as  they  have  ribs  working  in  the 
grooves,  which  they  are  compelled  to  follow.  By  the  weight  suspended 
from  the  vertical  rack,  any  irregular  motion  lliot  might  take  place  is  en- 
tirety obviated,  it  being  kept  firmly  n'orking  on  the  upper  surface  of 
'Sa.ii  curved  bar  it.  while  the  small  handle  is  provided  for  the  purpose  of 
raising  it  by  band  when  required.  There  are  two  tools  l\  Fig.  3,  filed  to 
one  of  the  four  dies  already  described  ;  tlie  one  farthest  from  the  frame  /  is 
for  rough  turning,  while  the  other  follows  it,  giving  the  finishing  cut. 

A  small  bearing  is  provided  on  the  bed  for  supporting  the  outer  end  of 
le  screw  j^  and  the  standards  are  well  secured  to  the  floor  by  bolts  for 
"QlKt  purpose. 

This  machine  is  capable  of  turning  and  finishing  three  barrels  per  hoar ; 
and  similnr  machines  tire  in  use  at  tlte  Hoval  Armoury  of  Enfield,  fitted  up 
Llovd  and  Co.,  and  twelve  of  them,  in  conjunction  with  a  com- 
f  mncliinery  fur  nmkiiig  musket^  arc  at  work  in  the  Imperial 


432  EXPLANATION  OF  THE  PLATES. 

Armourj  at  Constantinople,  constnicted  by  Messrs.  Bemue,  bendes  ma- 
chines for  tlie   French  Goverament  and  for  his  Highness  the  FmIm  of 

PLATE  XXIX.— Figs.  1  and  2. 
Portable  Hand  Drill,  by  Messrs.  Nasmtth,  Gaskbll,  and  Co. 

The  utility  of  the  little  machine  shewn  by  Figs.  1  and  2,  is  its  porlir 
bility ;  and  it  is  found  very  conycnicnt  in  drilling  holes  in  sach  pieces  of 
machinery  which  if  required  to  be  brought  under  a  larger  machine  would 
cause  great  delay  and  iuconyenience. 

The  frame  a  carries  the  upright  drilling  spindle  b^  the  top  of  which  is 
a  screw  e'  for  raising  it  by  the  handle  wheel  e  by  hand,  while  the  revolTing 
motion  is  communicated  to  it  by  the  two  small  bevel  wheels  for  conveying 
it  at  right  angles.  When  required  to  drill  a  hole  in  any  piece  of  machinery, 
it  is  first  of  all  set  in  its  proper  place ;  after  tliis  is  done,  the  handle  c,  or 
small  fly  wheel,  is  turned  round  for  working  the  drill,  and  by  a  slow  re- 
volving motion  given  to  the  upper  handle  ^,  the  drill,  while  working,  is 
made  gradually  to  descend.     It  can,  if  required,  be  secured  to  its  woxlc. 

Fig.  3. 
Foot  Duill,  by  Messrs.  Nasmyth,  Gaskell,  and  Co. 

The  diifcrencc  between  this  machine  and  those  described  by  Plates  XXX. 
and  XXXI.,  consists  in  the  mode  by  which  the  upright  drilling  spindle  is 
made  to  rise  and  fall.  In  those  referred  to,  this  is  made  self-acting, 
while  in  this  case  it  is  performed  by  the  pressure  of  the  foot  of  the  man 
superintending  the  drilling  of  the  hole. 

This  machine  is  driven  by  the  riggers  or  pulleys  ^,  the  one  running  loose, 
while  the  other  is  fixed  to  the  spindle  for  conveying  the  motion  by  means 
of  the  upper  and  lower  sets  of  pulleys  f,  by  which  the  speed  can  with 
great  ease  be  made  to  ^-ary  considerably ;  this  is  done  by  altering  the  posi- 
tion of  the  leather  strap  shewn  by  the  dotted  lines.  The  motion  is  then 
carried  at  right  angles,  to  the  drilling  spindle  by  Uie  bevel  wheels,  for  pro- 
ducing the  requisite  revolving  motion. 

A  moveable  table  ^,  for  carrying  the  work  to  be  drilled,  is  fixed  to  the 
frame  a,  in  which  it  slides,  and  can  be  raised  or  lowered  by  the  wheel  and 
screw  // ;  this  is  found  of  great  use,  as  the  size  of  work  may  very  mudi 
vary. 

By  means  of  the  footboard  y^  working  as  a  lever  on  its  fnlcmm,  the 
drilling  spindle  is  made  to  rise  and  faHl ;  the  pressure  of  the  foot  on  the 


EXPLANATION  OF  THE  PLATES.  433 

boBrdycauaing  therody  to  rise,  which  by  the  upper  lever  fixed  to  tlio  frame 
of  the  machine  depresses  the  spindle  d,  while  it  is  revolving ;  as  soon  bs 
the  pressure  is  taken  off,  tlie  counterbalance  weight/"  causes  the  drill  to 

» ascend  and  take  its  former  position,  where  it  is  kept  until  again  used. 


PLATE  XXX. 
Wall  Sidb  Drilling  Machine,  by  Mbssrh.  Nasmyti 


The  eeverol  parts  of  this  mnchinc  con»st  of  a  frame  fixed  to  the  side  of 
le  wall  of  the  building,  ngEin§t  which  arc  bolted  the  frames  ib,  for  carry- 
iitg  the  upright  drilling  bar  e,  put  in  motion  by  the  driving  pulleys  or  riggers 
c,  according  to  the  Hpccd  required,  which  is  regulated  by  the  diameter  of 
the  pulley  on  wbicli  tho  atrap  works.  Independent  of  this,  it  has  also  a 
double  motion,  obtained  by  the  two  spur  wheels  pad  pinions  d;  this  second 
motion  is  fully  explained  in  the  reference  to  the  Plates  XXXIII.  and 
XXXIII  A. 

This  machine  is  made  self-ncting  by  the  spur  wheels  and  pinions  // 
mniig  the  drilling  bar  as  they  revolve ;  the  njiper  port  of  the  screw  g  has  a 
cross  guide  h,  slidtug  up  and  down   between    the  two  upright  parallel 

Tho  table  k,  for  carrying  the  work,  is  made  to  slide  on  the  bedy,  similar 
to  Ae  bed  of  a  planing  machine,  one  of  its  sides  being  of  the  V  form,  while 
the  other  is  a  Hat  smooth  surface ;  this  table  is  advanced  by  a  chain  fixed 
to  it  at  one  end,  and  works  round  the  rollers  m. 

In  the  drawing,  a  small  cylinder  n  is  being  bored. 

This  machine  is  well  adapted  for  boring  holes  fur  the  pivots  of  engine 
and  oU  parallel  holes. 

PLATE  XXXI. 

I  DoDBLE  Pillar  Drill,  by  Messbs.  Nasmyth,  Gaskbll,  and  Co. 

I  The  principle  of  this  mochino  is  the  same  as  that  last  described,  witli  the 
f  a  fen-  of  its  parts,  which  are  of  larger  dimensions,  and  for 

The  fmroe-work  o  for  supporting  the  different  parts  of  the  machinery, 

Bts  on  two  upright  pillors ;  on  this  fmme  is  a  small  shaft  for  carrying  the 

hring  pulleys  b  and  pinion  c,  conveying  the  motion  to  the  upper  or  inlcr- 

iate  shaft  by  the  spur  wheel,  whence  it  is  taken  at  right  angles  to  the 

g  bar  by  the  bevel  wheels  rf;  the  bar  e  is  raised  and  lowered  by  the 

D  the  upper  jiart  of  this  bar,  which  screws  itself  up  in  a  nut ;  when 

uirad  to  bo  lowered,  the  handle  or  whcoiy  is  worked  ronnd  by  hand. 


434  EXPLANATION  OF  THE  PLATES. 

setting  in  motion  the  upper  spur  wheels,  thereby  commnnicatiiig  it  to  At 
upright  screw  and  drilling  bar. 

The  moveable  bed  or  table  hy  for  carrying  the  work,  travels  along  the  bed 
g  ;  it  has  two  motions,  the  one  at  right  angles  to  the  other,  and  by  the  long 
screws  and  handles  i  i  is  brought  in  any  convenient  position. 

Its  principal  adaptation  is  for  boring  the  holes  for  receiving  the  tabes  in 
locomotive  boiler  plates,  and  in  such  coses  when  any  number  of  holes  are 
required. 

PLATE  XXXII. 
Radial  Dhilling  Machine,  by  Messrs.  Benj.  Hick  and  Son,  Bolton. 


An  entirely  ditfcrcut  arrangement  of  a  drilling  machine  from  those 
rally  used,  may  be  seen  by  this  Plate ;  in  all  ordinary  cases,  the  drilling  bar 
or  spindle  is  stationary,  that  is  to  say,  it  has  no  lateral  motion,  being 
only  able  to  rise  and  fall  in  its  bearings ;  in  this  case,  the  whole  drilling 
tackle  is  made  to  slide  along  a  radial  bar  or  carriage,  whereby  the  drill  can 
be  brought  over  the  work  into  any  required  portion  within  the  limitB  pre- 
scribed by  the  radius  of  the  arm  b. 

The  arrangement  of  this  machine  consists  of  a  strong  upright  column  a» 
bolted  in  a  most  substantial  way  to  tlie  stone  foundations ;  a  screw  c  worid 
up  and  down  in  the  internal  part  of  this  column,  according  to  the  he%fat 
required  for  the  work ;  this  is  made  secure  when  raised  to  its  proper  posi- 
tion by  a  uut  d^  tightened  by  the  four  pins  on  its  circumference;  the  apper 
part  of  the  screw  lias  a  collar,  upon  which  the  nidial  bar  rests,  which  at  the 
same  time  is  capable  of  revolving  on  this  centre ;  a  carriage  for  supporting 
the  small  square  slmft,  the  bevel  \>'liec]s,  and  the  fast  and  loose  puDejB 
placed  horizontally,  is  fixed  to  the  upper  part  of  the  radial  bar  by  bolts 
and  nuts ;  on  the  other  extremity  of  the  shaft  y,  is  a  second  pair  of  bevel 
wheels,  for  conveying  the  motion  at  right  angles,  thus  causing  the  drilling 
bary  to  revolve,  on  the  lower  end  of  which  is  fixed  as  usual  the  drill  it. 
The  two  upright  sujiports  h  are  bolted  to  the  travelling  frame ;  these  carry 
the  apparatus  for  raising  and  lowering  the  drill,  which  consists  of  two  small 
chains  fixed  to  the  top  of  the  diilling  bar,  working  round  rollers,  and  also 
two  others  having  weights  suspended  from  them,  and  running  over  the  chain 
])ulleys,  that  with  the  large  weight  brings  tlie  drill  down,  while  it  is  drilling, 
which,  after  it  has  performed  its  duty,  is  disengaged  by  a  lever  and  rod,  ao 
that  its  weight  is  neutralized,  when  that  of  the  smaller  ball  comes  into 
o])eration  for  raising  the  drilling  bar  to  its  former  position.  On  the  lace  of 
the  radial  bar  is  a  rack  and  pinion  /,  which,  by  means  of  the  wheel  handle 
l\  causes  the  drilling  frame  to  slide  along  the  surface  of  tlie  bar  6,  whiob,  as 
already  stated,  is  regulated  by  the  position  required  for  the  drill,  while  tbe 


EXPLANATION    OF    THE    PLATES. 


435 


■qoara  epiudlo  or  sbntl  g  slides  through  the  wLocl,  nnd  also  the  bearing  fixed 
on  the  bar.  Th?  motion  Is  conininnicsted  by  a  leather  stm|)  on  the  fast 
pulley  e,  tranBmitling  it  through  the  bpve!  wheels  to  the  drilling  bar. 


PLATES  XXXIII.  AND  XXXIII.  a. 

Upright  Dhillino  and  Borino  Machine,  bv  Mh.  F.  Lewis, 
Manchester, 

In  most  drilling  machiaeB,  the  motion  ia  conveyed  at  once  to  the  npright 
I  bar,  by  fixing  the  driving  polleya  ou  it,  which  is  the  case  in  those  already  de- 

eribed,  os  will  be  seen  by  rcrerriiig  to  the  Plates ; 
[  mediate  shaft  or  spindle,  placed  horizoutally,  re 
[■■trap  on  the  pulleys  or  riggers  c,  and  is  conveyed  a 

r  of  small  bevel  wheels  k ;  a  double  motioi 
[  tnacbine  whereby  the  epceJ  can   be  much  altered, 


n  this  iustaiicc,  an  inter- 
«ives  the  motion  by  a 
,t  right  angles  bv  means 
1  is  provideil  to  this 
independently  of  the 
F  difierent  diameters  given  to  the  driving  pulleys.  The  mode  by  which  tbia 
alterauon  of  speed  is  obtained  is  thus :  there  ore  two  shafts,  one  having  b 
wheel  working  in  a  pinion,  and  the  other  a  pinion  working  in  a  wheel,  each 
pwr  being  of  the  some  diameters,  Fig.  2 ;  it  must  be  understood  that  the  pul- 
bys  e  and  the  pinion  shewn  in  Fig.  1,  run  loose  on  tho  shaft  on  which  Uio 
[  ftontwhceliskeyed,  andarethore  st-cnrcd.  Now  suppoang  a  slow  speed  bo 
LveqniTcd,  it  is  only  necessary  to  throw  in  gear  the  bnck  wheel  and  pinion, 
■  sliding  the  shall  in  a  groove  where  it  may  be  kept  fixed  by  a  pin  </", 
The  result  of  this  is,  that  the  pulleys,  running  loose  on  the  front 
I,  convey  the  motion  through  the  jiiuion  fixed  to  them  to  the  back 
iteel,  and  hence  through  the  bock  pinion  to  the  front  wheel,  conveying 
1  through  the  bevel  wheels  to  the  drilling  bar  e.  An  esamplc  will 
better  illustrate  this : — Let  it  be  supposed  tlmt  tlie  pulleys  and  pinion  are 
Buking  twenty  revolutions  per  minute;  the  wheels  being  IC  inches  in 
diameter,  and  the  pinions  6  inches  in  diameter ;  the  speed  of  the  bock  shaft 
wonld  be  1\  turns  per  minnle,  and  consequently  that  of  the  wheel  and  front 
abaft  would  be  reduced  Co  about  3^  turns,  thus  making  the  respective  velo- 
cities about  5  J  to  1.  When  a  <(uicker  speed  is  required,  the  back  shaft, 
wheel,  and  pinion  ore  disengaged,  (by  the  same  meaiis  it  was  engaged,  and 
secured  by  the  pin  cT',)  the  front  wheel  is  then  made  fast  [o  tlie  pul- 
leys by  a  screw  d'  on  its  face,  and  both  revolve  at  the  some  velocity.  Tho 
k  wheel  and  pinion  should  be  always  disengaged  previous  to  fixing  the 
(Ont  wheel  to  the  pulleys,  otherwise  great  damage  would  be  done  to  the 
JKth  of  the  wheels. 

e  self-ucting  moliou  in  this  machine  is  the  same  aa  those  generally 
I,  whidi  is  llirown  ofTnnd  on  by  the  small  clutch  k  and  lever  i,  imd  may 


43G  EXPLANATION    OF   THE   PLATEB. 

be  worked  by  hand  by  the  wheel  /;  the  screwy  works  in  tlie  niit^%  and 

by  the  small  spur  wheclsy^  raises  the  drill  bar. 

The  table  m  for  carrying  the  work  to  be  drilled  or  bored,  is  raised  or 
lowered  according  to  circumstances,  by  the  two  pinions  n  VForking  in  the 
racks  fixed  to  the  \v^\  plate ;  this  is  effected  by  the  handle  o,  wheel  and 
pinion  /?,  and  is  kept  from  returning  by  tlie  paul  and  ratchet  wheel  q  pro- 
vided for  that  purpose ;  the  plate  r  is  bolted  to  the  wall  through  four  snv^ 
The  frames  h  for  supporting  the  several  parts  of  the  machinery  have  snitsUe 
bearings  on  them  for  carrying  the  shafts  and  spindles ;  they  are  ivcll  secured 
to  the  wall  by  strong  bolts  and  nuts. 

This  machine  can  bore  from  lialf  an  inch  to  14  inches  in  diameter,  with 
a  depth  of  two  feet,  and  tlie  table  m  vnUl  admit  of  a  wheel  3  feet  8  inches 
in  diameter  being  placed  on  it. 

PLATE  XXXIV. 

Slotting  or  Key  oroovino  Machink,  ry  Messrs.  Nasmyth,  Oasrell, 

AND  Co. 

This  machine  is  of  a  very  simple  construction ;  the  motion  being  under- 
ground, and  conveyed  to  it  by  the  small  shafts  to  a  dog  or  crank  wheel  ^ 
giving  the  slotting  bar  a  reciprocating  motion  by  means  of  the  connectiiig 
rod  r,  to  which  it  is  fixed  ;  the  lower  part  of  the  slotting  bar  works  in  a 
cross  guide.  To  the  sliding  ])late  ^/,  the  wheel  e  to  bo  grooved  is  fixed ;  the 
cut  is  pcrfomu'd  by  the  descent  of  the  slotting  bar,  the  length  or  depth  of 
which  is  regulated  by  a  moveable  crank  ])in  on  the  face  of  the  plate  b»  It 
is  necessary  to  have  access  to  the  underground  work,  and  for  that  purpose 
the  stairs  f  are  shewn.  A  machine  u])on  this  principle  for  cutting  the 
teeth  of  wheels  M'as,  we  believe,  used  by  Messrs.  Bolton  and  Watt,  at  Soho. 


PLATE   XXXV. 
Slotting  Machine,  by  Messrs.  Sharp  and  Roberts,  of  Manchestbr. 

The  persj)ective  view  shewn  by  this  plate  represents  a  machine  for  slot- 
ting or  poring  small  work,  such  as  the  straps  of  connecting  rods,  or  cuttine 
the  key  grooves  of  wheels  of  small  diameters. 

The  table  /  for  carrying  the  work,  is  a  circular  plate  of  about  two  feet  in 
diameter.  It  has  two  horizontal  motions  at  right  angles  the  one  to  the 
other,  working  between  V's,  and  these  are  made  self-acting  by  means  of  the 
ratchet  wheels  o,  o,  worked  by  the  cam  d  on  the  main  driving  spindle  c, 
which,  08  it  revolves,  strikes  against  the  lever  k,  and  thereby  gives  them  a 


EXPLANATION    OF   THE    PLATES. 


4.37 


I 


I 


progrcBiiTe  motion ;  the  pauls  pp  aJvance  it  in  proportion  to  the  leverage 
i>r  distance  from  tlie  fulcnim  given  to  lever  i;  this  may  be  altered  at  pleasure 
by  clianging  tbe  position  of  the  rod  r.  The  Beif-acting  progregs  of  the 
table  /  is  thus  made  to  vary  from  ^\  to  -^^  of  an  inch  per  revolution  of  the 
entnk  wheel  g.  The  spiral  spring ./'  serves  to  put!  up  the  rod  after  it  has 
been  forced  down  by  the  cam  rf,  and  bring  back  the  paiils  for  a  new  stroke. 
The  use  of  the  other  pauls  is  to  prevent  the  ratchet  wheel  from  running 
bock  while  tliis  latter  operation  is  being  performed.  Another  ingenious  con- 
trivuncc  in  this  machine  consists  in  a  third  or  circular  motion  of  the  plate  /, 
which  turns  on  its  centre,  and  linving  its  circuraferencc  equally  divided  hy 
tfie  notches  m,  may  be  moved  ronnd  an  equal  angular  distance  at  a  time, 
■od  by  the  spring  or  catch  n  is  retained  in  a  Bxed  position  while  the 
tool  is  nt  work.  Tliis  arrangement  is  found  to  be  very  advanti^oiis  in 
cutting  the  key  grooves  of  wheels,  being  required  to  be  perfectly  equi- 
distant. 

The  riggers  /  arc  worked  by  a  strap  from  the  main  shaft,  and  next  to 
tfaem  is  hung  a  tly  wheel  e,  to  regulate  the  motion ;  these  and  the  emnk 
wheel  g  are  securely  keyed  to  the  spindle  c.  The  reciprocating  motion  is 
then  carried  to  the  slotting  bar  b,  hy  a  smal!  connecting  rod  h;  it,  are 
groove-pieces  fixed  to  the  frame  in  which  this  bar  works ;  the  length  of  the 
stroke  may  be  altered  by  slidiug  the  cnmk  pin  «,  in  a  groove  provided  on 
ibo  plate  _o,  but  its  longest  stroke  cannot  exceed  8  inches.  The  different 
parts  of  this  machine  are  fitted  to  the  frame  or  standard  a,  having  on  its  two 
inteniol  sides  V's,  in  which  the  table  frame  slides,  which  may  be  either  raised 
or  lowered  by  the  handle  underneath.  The  screw  handle  q  and  also  the 
huidle  on  the  ratchet  wheel,  serve  to  bring  back  the  tables  after  they  have 
advanced  hy  the  self-acting  apparatus.  The  action  of  this  machine  is 
to  that  of  the  morticing  machine  at  Portsmouth. 


PLATE  XXXVI. 


Machine  fob  cuttino  Kev  Ghooves  in  Wberls,  bv  Messrs, 
Nasmvth,  Gase^sll,  and  Co. 


Another  descri 
iffering  merely  in  detml  from 


iiotti 


ng  machine  is  represented  hy  this  Plate, 
last  described,  which  is  limited  to  the 
e  of  the  work  it  is  capable  of  performing  by  the  two  sides  of  the 
indard  frame  preventing  the  admission  of  large  wheels.  In  the  (iresent 
ichine,  a  wheel  of  any  size  can  be  grooved,  the  whole  of  the  machinery 
eing  uudcmeath  instead  of  above  the  table. 
The  four  small  colunms  a,  support  the  table  b,  on  which  there  is  a  bed 
for  ifac  alido  to  work  in ;  on  this,  the  dividing  pktc  m  b  placed,  having  iU 


438         EXPLANATION  OF  THE  PLATES. 

ciTcainfercncc  notched  in  a  similar  way  to  the  machine  deacribed  in  Fbte 
XXXV.  The  spindle  e,  is  driven  by  the  rigger  e,  and  baa  a  Hj  wlied  d^ 
to  regulate  its  motion ;  on  tlie  end  of  this  same  spindle  is  a  pinion,  wotkii^ 
in  the  crank  spur  wheel  y^  connecting  the  motion  to  the  slotting  bar  by  tbe 
rod  k ;  the  position  of  the  crank  pin  ^,  may  bo  altered  to  any  convenient 
length  of  stroke  by  sliding  it  along  the  groove  shewn  on  tbe  face  of  tbe  apor 
wheel  /.  Thus  a  reciprocating  motion  is  given  to  the  bar  t^  baring  on  Hi 
upper  end  the  tool  for  cutting  the  grooves,  represented  by  the  dotted  Ibifli 
on  the  large  spur  wheel  being  cut.  This  machine  is  made  self-acting  by 
eccentric  or  cam  on  the  spindle  y^  which  raises  and  lowers  the  small  1 
by  means  of  upright  rods  working  the  ratchet  wheels  jy  and  paola  t;  tbe 
ratchet  wheels  are  fixed  to  long  square  threaded  screws  (the  length  of  tbe 
bed),  and  work  through  a  nut  fixed  to  the  underside  of  the  sliding  tabk^ 
which  may  be  inclined  or  placed  at  an  angle  suitable  to  the  taper  reqnired 
for  the  key.  The  handle  /  prevents  the  circular  dividing  plate  us  from 
changing  its  position  while  the  groove  is  being  cut^  after  which  it  is  dis- 
engaged from  the  notch  by  hand,  till  it  meets  a  second  one,  eqnidistsntly 
divided. 

Between  the  columns  tlie  diagonal  stays  are  placed,  to  give  strengtb  and 
stability  to  the  machine,  there  is  also  a  cross  plate  running  from  one  side 
frame  to  the  other,  through  which  an  aperturo  is  made  to  allow  tbe  slotting 
bar  t  to  work  through. 

The  tool  on  the  top  of  the  slotting  bar  can  be  altered  at  pleasnre,  by  un- 
screwing the  small  screw  which  secures  it  in  its  place. 

It  is  from  the  advantages  derived  by  the  principle  of  this  machine,  wbicb 
admits  wheels  of  unlimited  diameters,  from  its  underneath  motion,  tbst 
Messrs.  Nasmyth,  Oaskell,  and  Co.  have  founded  their  patent 

PLATE  XXXVII. 

Large  Slotting  Machine,   by    Messrs.  Nasmyth,  Gaskbll,  and  Co. 

Among  the  variety  of  slotting  and  key-grooving  machines  already  do- 
scribed,  there  has  been  none  similar  to  this,  cither  in  the  general  arrange- 
ment of  its  parts,  or  its  capabilities  as  to  the  magnitude  of  the  work  it  is 
capable  of  slotting ;  it  is  altogether  a  much  larger  machine,  and  of  simple 
contrivance.  A  cmnk  is  shewn  in  the  drawing,  having  its  sides  pored  off 
by  the  tool ;  however,  it  is  not  solely  confined  to  cranks,  but  may  bo  used 
for  any  machinery  that  can  be  placed  within  the  limits  prescribed  by  tbo 
two  columns. 

The  arrangement  of  this  machine  consists  of  a  rectangular  finme  a. 
Figs.  1,2,  and  4,  upon  which  arc  placed  the  plummer  Uoclu  for  euryiiur 


EXPLANATION    OF   THE    PLATES.  139 

ibo  shaft  bearings;  tho  two  columns  o' o'  resting  on  tlie  top  of  the 
bed  b  sapport  this  upper  eiilabloture  frame  a.  Like  most  slotting  ma- 
chines, the  plate  for  carrying  tlic  worlt  lias  three  different  motions,  the  two 
first  at  right  angles  the  one  to  the  other,  while  the  third  consists  of  a  circu- 
lar motion,  which  is  required  when  the  work  to  be  operated  upon  is  drcu- 
iar.  TIic  first  is  longitudinally  along  the  bed  b ;  this  is  done  by  working 
the  ratchet  wheel  and  screw  e",  thus  drawing  the  slide  e ;  the  second  at 
right  angles  by  the  ratchet  wheel  and  screw  d"  working  the  slide  <^  in  a 
transverse  direction  ;  and  lastly  the  third,  which  con^sts  of  a  circular  plate 
having  on  its  circumference  the  worm  wheel  c,  made  to  revolve  on  its 
centre  by  working  the  worm  or  endless  screw  and  the  ratchet  wheel  c*.  The 
screws  ^  and  if  severally  work  in  nnts  fixed  to  the  under  sido  of  the  slides, 
one  of  which  may  bo  seen,  by  the  dotted  lines  shewn  by  Fig.  3  in  the  plan 


of  the  bed-plate.  Fig.  4  shi 
tool  or  slolilng  bar  i ;  the  n 
i  from  the  driving  Bboft,  is 
which  is  a  spur  wheel  and  pir 
to  the  large  spur  wheel  i,  t 


(pvih 


plan  of  the  driving  gear  fur  working  Uio 
being  conveyed  to  the  riggers  or  pulleys 
icatcd  to  another  set  of  pulleys  upon 
n,  conveying  it  through  an  intennediate  shall 
which  is  fixed  the  connecting  rod_/;  by  this 
connection  tlie  wheel  A  answers  Loth  this  purpose  and  that  of  the  crank, 
lying  the  alternate  or  up  and  down  motion  to  the  slotting  bar  k. 
different  arrangements  that  may  be  ^ven  to  the  wheels  and  driving  rig- 
pulleys  I,  independent  of  i!ic  double  motion,  similar  to  that  de- 
ibcd  by  Plate  XXXIII.  and  XXXIII.  a.,  a  great  variety  of  speeds  may 
bo  obtelned,  which  is  of  importance,  some  ports  of  machinery  requiring 
a  much  greater  velocity  than  others,  white  being  opemted  upon. 

The  hollow  bar  k  is  guided  at  one  end  by  passing  through  the  upper  part 
of  the  frame  o,  which  is  all  one  piece ;  this  may  be  seen  in  Figs.  I  and  4 ; 
and  the  length  of  the  stroke  may  be  altered  to  suit  the  work  by  clianging 
tlie  position  of  the  connecting  rod  _;'  on  the  slotting  bar  it,  sliding  it  in  a 
■iDove  provided  for  that  purpose. 

sThe  three  different  motions  already  described  are  rendered  self-acting  by 
k|nn  on  the  spur  wheel  h  striking  as  it  revolves  agmnst  the  lever  shewn  by 
the  dotted  lines  in  Fig.  1,  which  commimicate  with  the  levers,  rods,  and 
small  bevel  wheelsy,  and  ultimately  givenself-aclingniotion  to  the  different 
Handles  arc  placed  on  llic  three  ratchet  wheels,  for  working  them 


;.  2  shews  the  tool  /  in  the  a 
|r  which  the  third  motion  is  put  < 


t  of  paring  the  circulor  part  of  a  crank, 

1,  and  the  other  two  are  consequently 


e  frame  is  strongly  secured  to  the  stone  foundutioj: 
Iag-<kiuii  bolts. 


440  EXPLANATION   OF   THE   PLATES^ 

PLATE  XXXVII.  A. 
Slotting  and  Paring  Machine,  by  Mb.  F.  Lewis,  Mani 

The  machine  represented  hy  Figs.  1  and  2  of  this  Plate,  thoo^  different 
in  detail  from  those  preceding  it,  is  the  some  in  principle ;  this  neccM 
rily  must  be  the  cose,  where  either  the  operation  of  slotting,  paring,  or  key- 
grooving  is  to  be  performed,  which  severally  require  an  alternate  or  reci- 
procating motion,  whereby  the  introduction  of  the  crank  or  dog  wheel  is  in- 
dispensable. 

The  dra\\ing  shewn  by  this  Plate,  is  a  slotting  machine  of  great  power, 
by  which  work  of  considerable  size  may  be  operated  upon,  the  table  for 
carrying  the  work  being  about  3  feet  in  diameter,  while  the  stroke,  wbidi 
may  be  varied,  can  be  extended  to  2  feet  6  inches. 

In  the  side  elevation  will  be  seen  the  frame  a,  to  which  is  connected  the 
whole  of  the  machinery  for  working  the  different  parts,  according  to  the 
nature  of  the  work  required.  On  the  front  of  this  frame  are  two  fiices^ 
Fig.  2,  to  which  are  fixed  the  V  pieces,  two  of  which  can  bo  adjusted  hj 
the  set  screws,  according  to  the  wear  of  the  slide  g.  Tlie  speed  may  he 
altered  to  suit  the  proper  velocity  for  the  work,  by  changing  the  str^i  on 
the  riggers  6,  these  receiving  the  motion  from  the  main  shaft ;  the  fly  wheel 
regulating  that  motion,  without  which  it  would  be  liable  to  jerks,  and  mnch 
variation.  By  the  wheel  and  pinion  e/,  the  speed  of  the  spindle  on  which 
is  placed  the  crank  or  dog  wheel  ^,  is  much  reduced ;  the  short  rod  f  con- 
nects the  up  and  down  motion  to  the  slide  y,  for  carrying  the  tool  f ;  andhy 
loosening  the  screws  h  h^  any  tool  can  be  applied.  The  spindles  for  carrying 
both  the  wheel  and  pinion  dd  run  freely  in  bearings  fixed  to  the  frame  a. 
On  the  top  of  the  bedy,  the  sliding  table  k  is  moveable,  and  on  its  upper 
surface,  the  circular  table  /  slides  in  a  contrary  direction,  this  latter  table  has 
also  a  revolving  motion  on  its  centre,  by  working  the  worm  on  the  circum- 
ference of  the  worm  wheel,  wliich  is  done  by  turning  the  handle  shewn  for 
that  purpose.  A  self  acting  motion  is  given  to  this  machine,  by  a  pin  on 
the  rim  of  the  upper  spur  wheel  d  striking  as  it  revolves  against  the  lever 
seen  in  Fig.  2.  by  the  dotted  lines.  On  the  lower  end  of  Uus  lever,  is  a  panl 
for  advancing  the  ratchet  wheel,  the  boss  of  which  is  a  brass  nut,  which, 
as  it  turns,  screws  up  the  square-threaded  screw,  and  draws  with  it 
the  table  k  on  the  adjustable  bed  j.  At  a  convenient  distance  on  this  lever 
is  placed  a  small  rod,  connected  at  tlie  bottom  with  a  series  of  levers,  which 
being  put  in  motion,  work  the  slide  frame,  and  give  also  the  circular  motion 
connected  to  it  by  the  two  small  wheels  and  paul,  thus  making  the  three 
motions  connected  with  the  sliding  frames  for  carrying  tho  work  entirely 
self  acting,  which  is  also  the  case  in  those  machines  already  described 
by  the  preceding  Plates. 


EXPLANATION  OF  THE  PLATES.  MJ 

aperture  through 


Tbat  part  shewn  as  circultir  on  the  main  fmmc 
krhich  tiie  spindle  e  U  passuil. 


PLATE  XXXViri. 

ScBEw-CnTTisG  Machine,  bv  Mb.  Fox. 

Tlie  machinery  required  for  this  purpose  consists  of  a  bed  a  anpjiorted  by 

o  end  frames;  upon  these  are  fixod  the  requisite  hearings  or  plummer 

•oks,  i  and  d,  for  carrying  the  two  spindles  c  and  e, — the  latter  being 

driven  by  the  riggers  i,  which,  by  the  small  double  conical  friction  clutch _/, 

ore  mode  to  work  or  revolve  iu  opposite  directional,  thus  keeping  the  ma- 

^^  chine  in  constant  operation.    By  mcaus  of  the  spur-wheel  and  long  pinion  k, 

^H|4lic  motion  ia  conveyed  to  the  Upper  or  working  spindle  of  the   chuck  or 

^^Bbnx  ni,  for  holding  the  screw  or  bolt  to  be  cut,  while  passing  through  the 

^^Bfes  of  the  outer  frame,  (which  ore  tightened  by  the  Iiandic  and  screw  I 

^^Bressing  against   the  two   springs  _;',)    thus  working  itself  through    the 

^^Hfiea,  and  drawing  with  it  the  spindle  c  and  upper  wheel  h,  the  length 

^^'reqiiired.      On  the  bos  wi,  Fig.  2,    is  shewn   a  screw  and   handle  for 

the   purpose   of    tightening   or  slackening  the  screw  before  and  afWr 

the  opcradon  of  cutting.      The  machine  is   put   in   motion  by  turning 

^»^  Btuall  handle  and  eccentric  y,  ou  the  top  and  bottom  of  the  upright 

^^hods,  which  press  tlie  triction  clutch  against  cither  the  one  or  the  other 

^^hf  the  two  riggers,  whicli,  from  running  loose  on  the  spindle  e,  can  be  made 

I^^lo  revolve  in  either  direction. 

When  it  ia  required  to  tap  nuts,  they  are  placed  in  the  outer  head,  and 
Bccnrod  by  the  wheel  I,  which  in  the  operation  described  above  contained 
the  dies,  the  tap  is  then  placed  in  the  bos  m,  and  aa  It  revolves  cuts  tlie 
proper  threat)  in  the  nut. 

kThis  machine  may  also  be  used  as  an  horizontal  drilling  machine,  the 
<rk  to  be  drilled  being  secured  in  the  head.  Fig.  3,  and  the  drill  In  the  box 
the  screw  above  the  riggers  being  then  tightened,  presses  forward  the 
11,  which  as  it  advances  drills  the  required  hole. 

The  spindle  which  carries  the  box  m,  might  he  made  to  slide  independ- 
ently of  tlie  wheel  and  pinion  h  h,  simply  by  having  a  lon^tudinal  groove 
T  keyway  cut  on  its  circumference. 


PLATE  XXXVIII.  A. 
I  Labos  Screw -Cutting  Macrini!,  bv  Messrs.  Nasuvth,  Qabkbll, 

AND    Co. 


I  This  machine  is  a  contrivance  for  the  purpose  of  cutting  screws  of  large 
Samctero,  nnd  couiusts  merely  of  two  side  tVames  or  standards  for  carrying 


442         EXPLANATION  OF  THE  PLATES. 

the  sercnil  parts  of  the  required  machinery.  Unlike  moat  machnieB,  tUa  lias 
neither  heel  nor  tahle,  the  frames  hcing  connected  together  by  the  two  atnmg 
wrought  iron  holts  dd^  (which  also  answer  the  purpose  of  guides,)  and  the 
two  stretching  holts  for  steadying  the  lower  port  of  the  standarda,  the  upper 
part  of  the  standards  aa  liaving  hearings  in  which  the  driving  abaft  woifa 
hy  means  of  the  pulleys  />.  This  machine  has  a  double  motion,  ramihr  to 
that  described  in  the  drilling  and  boring  machine,  Plates  XXXIII.  and 
XXXIII.  A.,  which,  as  it  is  there  shewn,  is  obtained  by  two  pairs  of  spur 
wheels  and  pinions  c.  In  this  case,  when  the  quick  speed  is  required,  the  knrcr 
shaft,  pinion  and  wheel  is  slid  through  its  bearings  in  a  simihir  way  to  die 
second  motion  of  a  crane  or  crab,  a  projection  being  ^ven  to  the  end  of  die 
diaft  for  that  purpose ;  and  when  the  slow  motion  is  used,  the  lower  shaft 
is  again  brought  into  the  position  shewn  by  Fig.  1,  the  wheel  coming  against 
the  collar  on  the  upper  pinion,  and  the  collar  of  the  lower  pinion  against  the 
upper  wheel ;  these  collars  prevent  them  from  going  any  farther,  while  die 
moveable  stop,  Fig.  4,  keeps  the  lower  spindle  in  its  proper  position. 

The  chuck  f^  Figs.  1  and  2,  forms  part  of  tlie  shaft  for  canying  die 
pulleys ;  and  in  it  are  fixed  the  dies  for  cutting  the  thread  on  the  sdew, 
which  may  be  taken  out  at  pleasure,  and  others  put  in  their  place,  by 
loosening  the  set  screws  ^ ;  a  second  chuck  or  frame.  Fig.  5,  (for  aappoii- 
ing  the  screw,  at  its  head,  between  the  dies,)  slides  along  the  guide  bolto  dd^ 
as  the  thread  is  being  cut  on  the  screw ;  when  this  operation  is  fiidahed,  die 
round  headed  screw,  Fig.  5,  is  slackened,  and  consequently  the  screw  is 
liberated  and  another  put  in  its  place. 

In  cases  where  nuts  arc  to  be  tapped  it  is  only  necessary  to  reverse  the 
operation  by  putting  the  nut  in  the  chuck  f^  and  the  tap  in  that  marked  e, 
where  they  are  secured  as  already  stated  in  reference  to  the  screw. 

The  arrangement  of  this  machine  is  extremely  simple  in  constroction, 
and  it  is  capable  of  cutting  the  threads  of  screws  of  considerable  diameters. 

PLATE  XXXVIII.  B. 

BOLT-SCKEWING    MACHINE,    BY'    MeSSKS.    BeNJ.    HiCK    AND  SON,    BOLTON. 

The  object  of  this  machine  is  the  same  as  that  described  by  Plate 
XXXVIII.  A.,  and  its  arrangement  and  principle  are  in  cxcry  respect  si- 
milar, with  one  exception ;  instead  of  a  double  motion,  as  is  the  case 
in  that  above  referred  to,  this  has  a  backi^-ard  and  foni*ard  motion 
given  to  the  hollow  shaft,  which  is  thus  effected  : — there  are  three  driving 
pulleys,  c^  Cy  d;  when  the  strap  runs  on  r,  the  machme  is  put  in  motion  bj 
the  spur-wheel  and  pinion  c\  thus  communicating  it  to  the  chuck  /^  for 
holding  the  steel  dies  or  cutters  ff ;  as  soon  as  the  whole  length  of  the  acfew 
lias  been  cut,  the  macliine  is  required  to  be  reversed,  which  may  be  done 


EXPLANATION    OF   THE    PLATES. 


'  shnft,  tliiia  drawing  tlic 


by  running  llio  strap  from  the  pulley  c  on  to  that 
wlieel  d'  is  set  in  motion  by  a  pinion  on  the 
on  to  the  lower  driving  spindlu 
former  position  ;  and  kstly,  when  the  machine  n 
on  to  the  circmiiference  of  the  centre  pulley  c, 
spindle,  revolves  u-ithout  producing  any  effect  on 
machine.  The  upper  bolts  b,  for  steadying  the  t' 
for  the  sliding  frame.    These  and  the  other  parts,  e 


of  d,  whereby  the  internal 

cylinder,  revolving  freely 


w  back  t< 


;  rest,  the  strap  is  passed 
licL,  being  loose  on  the 
le  working  parts  of  the 
'  frnmcB,  form  alsw  guides 
I  already  sfdd,  m 


IS  the  machine  last  described,  which  renders  all  further  description  useless. 
PLATE  XXXrS. 

&ELF-ACTIMQ   NUT-CUTItNO   MacUINE,   BV   MbSSOS.    NASIiyiH, 

Qaskell,  and  Co. 


rtl 
The  machine  shown  by  this  Plate  is  supported  on  a  fiarao  nmilor  to  that 
described  by  Plate  XXXVI.,  where  the  table  or  beJ,^  is  fixed  to  the  four 
columns,  which  ore  much  strengtliencd  by  the  diagonal  crosses, 

•  The  spindle  driven  by  the  riggers  or  pulleys  e  from  the  nuun  shaft  runs 
freely  in  brasses  fitted  in  the  heads  dd;  on  one  end  of  this  spindle  the 
■teel  uutter  a  is  fixed ;  the  slide y*',  on  which  is  placed  the  dividing  plate  or 
Llock,  is  advanced  between  the  two  V's  bv  a  screw,  having  at  one  of  its 
extremities  a  ratchet  wheel  i,  worked  from  the  main  spindle  by  means 
of  a  small  strap,  the  two  small  bevel  wheels  conveying  the  motion 
at  right  angles,  thus  advancing  the  ratchet  wheel,  which,  as  it  turns, 
vorka  (he  screw  through  a  nnt  fixed  to  the  under  side  of  the  slide  /', 
thereby  making  it  self-acting ;  _/  is  a  small  handle  for  the  purpose  of  bring- 
ing bodt  the  slide  by  hand,  after  it  has  performed  its  work.  By  this  ma- 
chine a  very  great  saving  of  time  is  effected  in  planing  or  siding  the  faces 
of  nuts,  witli  the  utmost  accuracy,  and  with  an  almost  incredible  saving  of 
time,  for  in  ordinary  cases  the  work  performed  by  this  machine  was  en- 
tirely done  by  cliipping  aud  filing.  The  mode  of  working  it  is  simply 
by  fixing  the  nut  as  it  comes  from  the  forge  on  a  mandril,  aud  tho 
(iktter  in  a  hole  on  the  block  A,  where  it  is  securely  fastened  by  a  small  nut 
re,  it  is  then  advanced  to  the  cutler  by  the  slldey,  where  it  receives  a 
ectly  smooth  face  ;  the  block  is  then  disengaged  from  the  handle  c  in 
0  notch  y,  and  made  to  revolve  till  it  comes  to  the  following  notch,  where 
\  u  agiun  secured.  The  block  is  divided  into  six  and  eight  equidistant 
portions  suitable  to  either  square  or  six-»ided  nuts. 
The  small  tank  I  contains  water,  kept  constantly  falling  on  the  cutter  a, 
T  the  purpose  of  keeping  it  cool. 

The  spindle  upon  which  the  nnls  ace  to  be  cut  may  be  suited  to  any 
s  isliewn  by  tho  spindle  tn. 


414  EXPLANATION  OF  THE  PLATES. 


PLATE  XL. 

Machine  for  CiTTiNr.  the  Teeth  of  Wheels,  bt  Mb.  F.  Lewis, 

Manchester. 

Bv  this  machine,  wheels  of  the  following  description  maj  be  cat:  lrt» 
common  spur- wheels ;  2n(Uy,  conical  or  bevel  wheels ;  and  3dly,  worm- 
wheels. 

a  a  represent  the  tr^'o  side  frnmes,  the  one  of  a  V  shape  for  the 
slide  ^  to  work  on,  while  the  other  is  a  flat  smooth  snr&ce.  The  mafthmeiy 
supported  by  these  frames  may  be  divided  under  two  different  heads,  tie.  : 
I  St,  that  required  for  giving  the  revolving  motion  to  the  wheel  to  be  col; 
and  2ndlv,  that  which  is  requisite  for  working  the  cutter  in  the  Tarioaa 
positions  it  assumes. 

1st.  Machinery  for  turning  the  wheel.  The  spindle  b  has  on  one  ex- 
tremity the  handle  b\  which  works  round  the  circumference  or  rim  of  a 
plain  wheel,  shewn  bv  the  dotted  lines ;  this  wheel  or  rim  has  on  its  cir- 
cumference two  notches^  equal  to  the  \i-idth  of  the  handle  b\  in  one  of  wUdi 
it  falls,  and  is  there  kept  fixed  during  the  operation  of  catting  one  tooA. 
On  the  opposite  end  of  this  spindle  is  a  small  spur  change- wheel  e,  working 
through  a  second  or  intermediate  one,  that  fixed  to  the  H'onn  spindle^ 
the  worm  d  then  conveys  the  motion  through  the  worm-wheel  e  to  the 
upright  spindle /i  on  the  upper  end  of  which  is  placed  the  wheel  to  be  cat 
This  spindle  revolves  in  brass  bearings  fitted  to  the  end  frame. 

The  change- wheels  and  pinions  ccc  may  be  altered  to  regulate  the  speed 
and  consequently  number  of  tcech  and  pitch  given  to  any  required  wheel, 
this,  as  will  be  seen,  is  very  easily  done  by  unscrewing  the  screws  which 
connect  the  bearing  pieces  of  the  two  pinions  and  intermediate  wheel  to  the 
side  frame,  when  wheels  of  ditferent  diameters  can  be  put  in  their  places  ac- 
cording to  the  required  motion.  This  at  once  regulates  the  distances  of  the 
bearing  pieces,  after  which  they  must  be  well  secured  to  the  frame  by  again 
tightening  the  nuts. 

2dly.  Machinery  for  working  the  cutter  ;/i,  consists  of  a  cross  slide  ^  tra- 
velling on  the  top  of  the  frames  a,  a^  already  described ;  this  is  moved  back- 
wards and  forwards  by  a  screw  working  in  a  nut,  and  a  small  handle,  whidi 
is  not  >een  in  the  drawing ;  on  this  slide  the  frame  with  the  two  headstocks 
/<f  h  is  securely  screwed  by  a  strong  bolt  and  nut ;  I'T  are  the  centres  by  which 
the  frame  i  i  is  fixed  to  that  Ia.st  described,  having  also  two  heads  Bunilar  to 
those  marked  /i  h  ;  on  the  back  of  this  frame  1 1,  is  a  bolt  and  nat  wUdi 
becomes  a  centre  for  the  sliding  frame  it,  and  by  which  it  is  fixed  to  it.  The 
following  motions  are  the  result  of  these  different  combinations:  IsL  Tlw 
frame  ^  is  advanced  by  the  screw.     2dly.  The  two  heads  fonnii^tfaeheMt- 


Iiangf 


EXPLANATION    OF   THE    I'l.ATES.  'i-iS 

stock  Frame  h  can  be  moved  horizontally  on  the  frame  ff  either  to  the  right 

or  left  on  its  centre,      3illy.   In  the  sanae  way  the  frame  ti  can  he  mode  to 

Iiang  forward  or  fall  bock  on  its  two  centres  i'i'.  And+thly.  That  markedy 

■s  a  position  either  te  the  right  or  left  on  its  centre,  while  the  alide  i, 

in  fact  carries  the  cutter  and  its  necessary  connections,  is  made  to  rise 

d  fall  by  a  small  pinion  working  in  the  rack  ^  and  receiving  iis  motion 

by  the  handle  k",  there  ore  two  V  pieces  ou  the  fntaiBj,  in  whicU  the  slide 

works  up  and  down  ;  these  are  filed  by  four  screws.     The  pulley  /  receives 

^^t8  motion  from  the  main  shaft  by  a  hand,  and  conveys  it  to  the  cutter  by  two 

^■■^  of  wheels  and  pinions,  by  which  the  speed  of  the  cutter  m  is  regulated  ; 

^^b  the  slide  it  is  filed  the  bearing  pieces  for  the  three  spindles ;  o  and  p  re- 

^^besent  an  adjusting  screw  and  index,  for  regulating  to  very  great  accuracy 

^Hh>  position  of  the  cutter  m,  which  may  eoaily  be  taken  out  and  replaced  by 

^Hniother  by  unscrewing  the  small  screw  n. 

I  The  frame  o  a  is  firmly  stayed  hy  bolts  and  nuts  shewn  on  its  side*. 

It  must  not  be  forgotten  that  when  the  headatock  frames  A  and  J  have 
assumed  their  new  positions  dcscrihed  by  the  second  and  fourth  motions,  there 
is  fixed  on  both  the  sides  of  the  screws  or  holts  upon  which  they  turn,  two 
smaller  holts  working  in  slols  or  grooves,  by  which  they  are  firmly  bolted, 
'  otherwise  the  frames  might  slip,  and  this  would  cause  great  damage  to  the 
Drk  being  operated  upon. 
1 ,  Any  one  of  the  different  positions  given  to  these  frames  may  be  uhtained 

lependently  of  the  others,  or  they  might  he  need  all  together. 
L  The  following  description  will  shew  the  mode  by  which  this  machine  is 
cn-lced  for  the  different  kinds  of  wheels : — 

s  present  position  it  is  regulated  for  cutting  the  teeth  of  a  common 

r  wheel,  which  is  securely  fixed  to  the  top  of  the  upright  spindle /"ond 

e  to  revolve  as  already  described  hy  tiie  handle  b',  which  works  from 

Utcb  to  notch  for  every  tooth,  the  change-wheels  ccc  being  regulated  to 

the  retjuired  number ;  hy  this  operation  of  the  handle,  the  distance  moved  bv 

the  wheel  to  be  init  is  always  the  same,  thereby  ensuring  the  utmost  accu- 

^^^y,     Tbe  whole  of  the  machinery  resting  on  the  frame  g  is  advanced  till 

^Hb  cutter  comes  close  to  the  wheel ;  the  shding  fnune  it,  to  which  is  con- 

^Hfpcted  the  cutter,  (which  is  now  put  id  motion  by  the  pulley  and  small  spar 

^^rhtsels,)  is  lowered  by  the  handle  working  the  rack  and  pinion  k'  till  tt 

comes  in  contact  with  the  tooth  to  be  cut ;  after  the  porfonuancc  of  thin 

operation  the  frame  p  is  drawn  back  by  a  screw  and  handle  already  de- 

libcd,  when  the  nlieel  is  again  made  to  revolve  the  distance  of  one  tootli, 

i  the  operation  of  the  cutter  repeated. 

In  the  case  of  a  bevel-wheel,  the  working  of  the  machine  is  similar,  the 
Y  difference  being  the  position  assumed  hy  the  cutter-frame,  whi(^  is 
B  to  incline  forward  at  any  angle  suited  to  tlio  bevel  by  llic  third  motion 


446  EXPLANATION  OF  THE  PLATES. 

described;  this  is  again  altered  when  it  is  required  to  cut  the  teeth 
of  a  worm-wheel,  simikr  to  that  shewn  in  this  Plate  by  e,  where  the  teeth  are 
seen  at  the  angle  of  the  worm ;  for  this  purpose  the  machine  assomes  the 
position  described  bj  the  fourth  motion ;  and  lastly,  i^hen  used  to  cut  the 
teeth  of  skew  berel- wheels  by  the  second  and  third  motions  combined. 

It  is  unnecessary  to  describe  the  utility  of  this  machine ,  or  its  acca- 
racy,  when  small  wheels  with  an  extremely  fine  pitch  are  required.  One 
has  now  been  in  use  for  some  length  of  time  at  the  Bank  of  England,  where 
it  is  found  to  be  very  useful  in  cutting  the  teeth  of  the  small  wheels  re- 
quired in  that  complicated  and  ingenious  machine  for  marking  the  numbers 
on  the  bank  notes,  the  numbers  changing  as  fast  as  a  man  can  feed  the 
machine. 

The  following  Plate  shews  a  larger  machine  of  the  same  description,  which 
may  be  better  understood,  and  has  the  same  letters  of  reference. 


PLATE  XL.  A. 

Machine  for  cutting  the  Teeth  of  large  Metal  Wheels, 

BY  Mr.  F.  Lewis,  Manchester. 

This  Plate  represents  two  geometrical  views  of  the  last  described  ma- 
chine, by  which  the  different  motions  vrill  be  understood  with   greater 
facility.     It  is  in  every  respect  similar  as  to  its  working  parts,  with  an 
additional  self-acting  motion  for  working  the  cutter,  which  operation  in  the 
last  case  was  performed  by  hand.     It  is  also  adapted  for  cutting  the  teeth 
of  much  larger  wheels  from  the  length  of  the  bed  or  frame  o,  upon  which 
the  cutter  frames  slide.     The  letters  of  reference  and  description  are  the 
same   as   those  described  in  Plate  XL. ;   consequently  a  repetition   ^ill 
be  useless,  it  will  only  be  necessary  to  describe  its  additions,  consisting  of  the 
self-acting  motion  obtained  by  the  bevel  wheels  s  communicating  the  motion 
of  the  driving  pulley  /  to  the  upright  spindle  f,  on  the  top  of  which  is  the 
spur  wheel  and  pinion  working  the  worm  w,  and  consequently  the  worm 
wheel  w'  fixed  on  the  same  spindle  as  the   handle  k^'  and  pinion  k\  thus 
raising  the  cutter  frame  ^,  on  the  back  of  which  is  fixed  the  rack.     The 
worm  wheel  ?/  is  seen  in  Fig.  2  by  a  dotted  line  behind  the  spindle  f.  The 
handle  X:"  serves  to  raise  the  frame  k  by  hand  after  the  tooth  is  cut  by  the 
self-acting  motion,  which  is  found  to  be  a  great  improvement  in  the  capa- 
bilities of  this  machine.     The  two  side  frames  a  a  are  connected  together 
bv  the  stretching  bars  v  provided  for  that  purpose. 

From  the  great  diameters  of  the  wheels  this  machine  is  capable  of  carrying, 
which  are  fixed  to  the  face  plate  o  by  the  bolts  and  nuts  o'o\  it  is  evident 
that  were  it  not  for  the  adjusting  screw  or  stay  /),  a  considerable  d^ree  of 


EXPLANATION  OF  THE  PLATES.  447 

motion  would  be  felt  at  that  part  of  the  circumference  where  the  tool  is 
operating,  and  great  inaccuracy  would  be  the  result ;  by  the  stay  /?,  this  evil 
effect  is  entirely  obviated,  as  it  may  be  adjusted  to  any  required  height. 

Wheels  of  the  follo^ving  description  and  sizes  may  be  considered  within 
the  limits  of  this  machine :  spur,  bevel,  worm,  and  skew  bevel  wheels,  five 
feet  diameter  in  iron,  and  10  feet  in  wood,  having  a  breadth  of  14  inches 
with  any  pitch  or  number  of  teeth. 


PLATE  XLL 

Machine  for  cutting  the  Teeth  op  Wooden  Wheels  Models  or 
Patterns,  ani^also  those  op  Iron,  by  Messrs.  Nasmyth,  Gaskell, 
AND  Co. 

This  machine  is  beautifully  drawn  in  perspective  with  the  usual  felicity 
of  Mr.  Nasmyth,  and  intended  for  the  same  purpose  as  those  described 
by  Plates  XL.  and  XL.  a.  Instead  of  the  machinery  being  fixed  to  a  table, 
it  is  here  made  to  slide  on  the  bed  6,  supported  at  both  ends  by  the  stand- 
ards aa. 

It  is  driven  by  the  leather  strap  c  on  the  rigger,  which  receives  its  mo- 
tion from  the  main  shaft.  The  bevel  mitre  wheels  d  then  convey  it  at  right 
angles  to  the  large  band  pulley^  whence  it  is  carried  to  the  small  pul- 
ley fixed  on  the  cutter  spindle^,  adjustable  by  the  set  screws/  both  at  top 
and  bottom.  The  spindle  e,  and  also  the  driving  spindle  upon  which  the 
rigger  is  fixed,  are  fitted  to  carriages  bolted  to  the  bed  b.  The  small  pul- 
ley /  is  for  the  purpose  of  keeping  the  band  /  tight,  as  the  position  of  the 
cutter  frame  alters ;  it  consists  of  a  weight  hung  over  a  tightening  pulley, 
the  weight  /  falling  as  the  bandy* slackens;  the  small  column  for  carrying 
the  pulley  being  fixed  on  the  slide  ^,  wliich  can  be  moved  the  whole  length 
of  the  bed  b ;  on  this  slide  is  shewn  a  circular  plate,  having  a  centre  on 
which  the  upright  frame  h  is  made  to  turn.  By  means  of  the  wheel 
and  handle  ib,  the  cutter  frame  t  can  be  raised  or  lowered  in  the  frame  last 
described,  by  loosening  the  two  nuts  shewn  on  the  back.  The  strength  of 
the  frame  h  is  much  increased  by  the  rib  shewn  in  the  drawing. 

It  is  obvious  from  the  foregoing  description  that  by  this  arrangement  the 
following  motions  are  obtained :  1st,  a  longitudinal  motion  along  the  bed ; 
2dly,  a  circular  motion ;  3dly,  a  transverse  motion,  which  is  required  for 
cutting  the  whole  width  of  the  tooth,  the  frame  being  worked  by  the  screw  A' 
and  handle  hf^ ;  and  lastly,  the  motion  necessary  to  adjust  the  position  of 
the  cutter  to  the  centre  of  the  wheels  whose  diameters  vary ;  this  is  per- 
fonned  by  the  handle  k. 

The  dotted  line  shewn  in  the  drawing  represents  a  very  large  spur  wheel 

o  o  2 


448  EXPLANATION    OF   THE    PLATES. 

pattern  of  mBlio<rany  being  cut ;  it  is  securely  fixed  to  the  diiidc  plate 
on  tlio  sliaft  Oy  which  is  made  to  revolve  in  the  two  heads  hj  ■ 
wheel  on  the  same  sliaft,  worked  by  the  worm  n  and  handle  for  that  pnrpowL 

The  mode  of  working  thi.s  machine  is  almost  similar  to  those  hwt  de- 
Rcrihed ;  the  wheel  being  accurately  clucked,  the  cutter  fraine  is  adyaneed 
along  the  bed  till  it  approaches  the  wheel,  when  the  handle  and  screw  If 
being  turned,  will  bring  the  cutter  to  the  surface  of  the  teeth.  This 
done,  the  handle  and  screw  //'  are  moved  in  a  contrary  direction,  till  the 
wheel,  being  cut,  revolves  the  distance  of  one  tooth,  by  the  smaU  handle 
and  worm  w. 

The  frame  p  in  the  adjoining  figure  may  be  put  in  the  place  of  that 
shewn  by  /,  when  the  machine  is  required  to  cut  the  teeth  of  iron  wheels, 
in  which  case  a  quicker  velocity  is  required,  as  will  be  seen  by  the  band 
wheel  and  also  the  spur  wheels  and  pinions;  the  spindle  for  connecting 
these  are  all  adjusted  by  small  set  screws. 

PLATE  XLLa. 
Machine  for  cutting  the  Teeth  of  Wheels,  by  M.  M.  Olavst. 


In  the  year  1 8»39  a  patent  was  taken  out  by  the  inventors  of  this 
cliiiie,  for  the  pur])ose  of  cutting  the  teeth  of  iron  and  wooden  wheelsi 
f'itlior  spur  or  hovel,  by  mechanical  nicnns.  It  is,  as  will  be  seen  by  the 
Platr,  alto;rt*tlior  on  a  ditforfiit  principle  from  those  described,  which  are 
certainly  wry  superior  in  capabilities. 

In  the  nrraiigenient  of  the  ])lan  and  section.  Figs.  1  and  2,  the  cast  iron 
table  a  is  supposed  to  he  su])ported  on  suitable  standards  for  that  purpose. 
On  this  table  or  bed  are  fixed  two  ]ilat«'s  lib^  with  bearings  for  the  spindles 
or  cylindrical  bars  cr ;  a  cap  is  fitted  to  these  bearings  and  secured  to  each 
of  tlieni  by  two  screws.  The  two  cylindrical  bars  are  united  together  at 
their  extremities  by  the  two  connecting  links  dtl;  these  bars  are  made 
square  in  the  middle  part,  and  the  two  V  pieces  it  are  fixed  to  this  square 
])art  of  the  bars  by  screws,  in  which  the  rest  //  for  carrying  the  tool  n  is 
made  to  slide  backwards  and  forwards;  it  is  adjustable  by  screws.  On  the 
s(piare  ])art  of  the  smaller  spindle  is  fitted  a  frame  in  which  a  carve  k  is 
dropi)e(l,  while  the  steel  sj>ring  /  presses  against  a  pin,  and  consequently 
the  curve  X*  against  a  screw y,  directing  the  tool  to  tlie  required  shape  of  the 
tooth.  This  i*<  <lone  by  the  bar  ^,  Fig.  2,  on  one  extremity  of  which  the 
tool  is  fixed,  and  the  screw  J  at  the  other.  By  a  contrivance  in  this  ma- 
chine, the  advance  of  the  tool  is  thus  made  self-acting;  two  Btofm  pp  are  ad* 
instable  in  the  mortices  Figs.  ],  2,  and  3,  and  are  regulated  according  to  the 
j<Jvfnec  n>qnired,  while  the  long  square-threaded  screw  m  works  in  a  hosi  on 


iron 


EXPLANATION    OF    THE    I'LATES.  449 

the  fnune  in  which  the  curve  k  is  situate,  its  other  extremity  being  tixed  to 
the  frame  o'  for  carrying  the  ratchet  wheel  o,  prevented  from  turning  cither 
one  w-ajf  or  lie  other  by  the  springs  or  paul«  jy. 

Tliis  machine  is  thus  put  in  motion ;  [he  wheel  to  be  cut  being  6xed  on 
its  projier  centre,  which  in  tliis  case  differs  from  those  already  described, 
^m  its  hciiig  secured  to  a  part  of  the  machine  itself.  An  olleniate  rao- 
is  conveyed  either  by  a  crank  or  other  suitable  means  to  the  rodey, 
conse<inently  the  two  cylindrical  hars  slide  in  their  bearings  bbbb, 
iwing  along  with  them  the  cutter  on  the  surface  of  the  tooth,  the  form  of 
which  is  regulated,  as  before  mentioned,  by  the  curve  k;  with  these  parts 
tlic  screw  m  to  which  it  is  attached,  and  also  the  ratchet  frame  and  ratchet 
wheel  are  moved ;  till  the  lower  part  of  thermmeo'.  Fig.  3,  comes  in  contact 
with  the  stop /J,  against  which  it  strikes,  thereby  throwing  it  from  its  verti- 
cal position  to  that  of  an  angle,  carrying  with  it  the  ratchet  wheel  to  which 
it  is  6xed,  thus  working  the  snuure- threaded  screw  and  ultimately  ad- 
vancing tlie  tool  n ;  the  opposite  stop  /*  is  so  regulated  as  to  bring  it  to  its 
original  vertical  position  when  in  its  turn  it  strikes  against  it.  By  the 
handle  m'  the  tool  is  brought  biitik  by  hand. 

Figs.  5  and  6  shew  the  adaptation  of  this  machine  for  the  imrpose  of 
cutting  the. teeth  of  conical  or  bevel  wheels;  consisting  of  an  adjustable  or 
jding  centre  pin  in  a  long  mortice  fixed  to  iIik  bod. 
From  the  above  description,  it  la  doubtful  whether  this  mnehine  has  ever 
practice  or  not,  owing  to  its  evident  imperfection  in  producing 
ly  thing  like  accuracy  of  workmanship. 

PLATE  XLII. 

iBTicAL  BoR/No  Machine,  by  Mbssb3.  Nasuvth,  Gabkell,  and  Co. 

\  The  many  advantages  derived  by  this  arrangement  of  a  vertical  boring 
machine  over  those  where  the  work  is  placed  in  an  horizontal  position,  may 
perhaps  be  unknown  to  persons  unocqnainted  with  the  general  character  of 
machinery;  it  may  not  be  unadvisablc  to  point  out  a  few  of  the  principal 
features  shewing  the  suiioriority  of  this  machine. 

lu  the  lirst  place,  tlie  arrangements  of  its  ports;  the  manner  in  which 
the  cylinder  is  placed,  namely,  its  vertical  poaitiun,  thereby  doing  airay  en- 
■ely  with  all  the  injurious  effects  produced  by  the  weight  of  the  body 
I  planed  or  bored;  thus  obviating  all  tendency  to  distort  Its  figure, 
bch  is  the  case  whcrr  the  operation  is  performed  by  the  horisoutal  sys- 
I,  and  where  the  sides  are  bulged  out  from  the  weight  of  the  upper  part; 
h  VMy  he  better  understood  bv  forming  a  cylinder  of  tliin  paper,  which 
Q  he  found  to  uiden  in  the  middle  and  assume  un  oval  form  from  its  own 

p«isli.. 


150  EXPLANATION  OF  THE  PLATES. 

Tliis  nltcration  of  fonn  is  foutid  to  be  quite  sensible  wben  the  cylinden 
urc  of  large  diameters.  Another  great  ad\*antage  of  this  system  of  verticd 
boring,  is  avoiding  all  risk  of  flexure  in  the  boriug  bar,  upon  which  the 
cntter  wheel  or  head  is  fixed  for  carr}'ing  the  boring  tools;  this  bar  his  « 
tendency  to  bend  down  in  the  centre  to  a  curve,  instead  of  keeping  a  per- 
fectly straiglit  line,  transferring  the  figure  assumed  by  the  bar  to  the  sorfaee 
of  the  cylinder ;  but  this  will  much  depend  both  on  its  length  and  diameter. 

Another  advantage  of  this  machine  is,  that  the  cutters  are  kept  dear  of 
the  borings,  which  fall  to  the  bottom  of  the  cylinder  as  fast  as  they  are  cnt. 
By  tliis  superior  arrangement  all  these  objections  are  entirely  remoTed, 
thus  avoiding  all  the  tendency  gravity  has  in  altering  the  tmeneas  of  the 
cylinder  or  the  bar ;  added  to  these,  the  power  requisite  to  bore  the  cylinder 
is  found  to  be  much  less  than  in  those  placed  horizontally,  a  very  desirable 
object  in  a  large  establishment. 

A  short  description  of  its  several  parts  will  enable  the  reader  more  foUj 
to  understand  the  ad^intages  already  alluded  to. 

Fig.  1  represents  a  cross  section  of  this  machine,  and  Fig.  4  a  plan 
shewing  its  position  in  a  comer  of  the  building  where  it  is  placed.  In 
these  two  ^-iews  it  will  be  seen  that  the  driving  part  of  the  machinery  is 
situated  below  the  ground  line  on  suitable  strong  foundations,  in  which  it  is 
inclosed.  These  parts  arc  rendered  accessible  by  the  steps  /,  which  are 
found  to  be  neccssar}'  in  cases  where  the  machinery  is  likely  to  get  ont  of 
cirdcr,  a  j)rociiution  never  to  be  neglected. 

The  two  ri^ircrs  XX"  receive  their  motion  from  the  main  shaft  by  means 
of  a  leather  stnip,  one  of  these  runs  loose  on  the  shaft,  and  the  strap  is 
iliroun  on  it  when  the  niaeliinc  is  not  at  work ;  this  is  done  at  pleasure 
with  the  greatest  possible  facility  ;  by  a  bevel  wheel  and  pinion^  it  is  then 
conveyed  through  the  shaft  /  to  the  endless  worm  ?/,  working  in  a  large 
worm  wheel  o,  which  is  fixed  on  the  great  vertical  boring  bar  a,  whereby 
a  verv  easy  motion  is  obtained,  and  all  jerks  avoided.  It  will  be  seen  by 
the  series  of  wheels  in  Fig.  4.  how  much  the  speed  of  the  boring  bar  is 
reduced.  The  shaft  /  is  placed  at  an  angle,  and  works  in  a  bearing  or 
]>hnnnier  Mock  and  a  step  //,  both  of  these  being  made  of  brass. 

The  vertical  bar  is  made  in  two  parts  a  and  c,  the  upper  one  a  for  cany- 
in^'  the  cutter  head  or  borin<r  wheel  r,  while  to  the  lower  one  is  connected 
the  driving  apparatus ;  they  are  coupled  together  by  the  upper  one  resting, 
as  is  shewn  in  Fig.  3,  in  a  socket  on  the  top  of  the  lower  one ;  a  steel  key 
/is  then  driven  in,  which  entirely  prevents  it  from  turning;  the  toe  of  the 
l)ar  r  rests  in  a  step  or  socket  shewn  by  Fig.  5  ;  the  entire  weight  of  this 
bar  and  its  appendages  is  thrown  on  the  hardened  cast  steel  disks  «,  which 
are  constantly  ke])t  supplied  with  oil.  Botli  extremities  of  the  bar  e  are 
rendered  adjustable  to  the  greatest  possible  accuracy  by  means. of  the  anall 


EXPLANATION    OF    THE    PLATES. 


451 


let  BcrewB  yy.  Figs.  3  and  5,  which,  by  heing  tightened,  press  against  the 
micttl  brass  Begtnents,  the  upper  one  forming  part  of  the  great  base  or 
»ot  plate  i,  which  is  niHterially  strengthened  by  mx  strong  rihs  on  its 
nnJer  wde,  Tlie  cross  beam  _?  is  well  fitted  to  the  sockeU  /,  built  into 
e  wall  of  the  building,  where  they  are  boiled  by  strong  bolts.  Figs.  1,  2, 
d  4:     It  has  an  addidonal  stay  in  the  bolt  n. 

There  are  four  standards  or  supports,  dd.  Fig.  1,  for  carrying  the  cylin- 
itr  to  be  bored,  which  can  be  altered  to  any  convenient  positioii  by  un- 
screwing the  bolts  which  fix  them  to  the  base  plate.     After  the  cylinder 
been  properly  placed  in  its  right  position,  it  is  fixed  to  these  supports 
by  clamps  e  and  bolts ;  and  thns  rendered  quite  immoveable. 

In  the  boring  bar  a  is  a  deep  socket  m.  Fig.  1,  which  alloy's  the  bar  to 
■lide  ap  and  down  by  means  of  the  screw  p  and  the  nut  /;  upon  the 
lower  aide  of  this  socket  is  a  flange  m,  upon  which  the  cutter  head  or 
wheel  r  rests,  receiving  its  motion  from  the  bar  by  means  of  a  nut, 
■nawering  both  the  pnrpose  of  nut  and  key.  By  the  dilTerent  arrange- 
ments of  the  sun  and  planet  motion  of  the  wheels  on  the  upper  part  of 
ibe  bar,  any  degree  of  motion  can  he  given  to  the  screw  for  the  descent  of 
the  cutter  wheel.  After  the  cylinder  has  been  once  bored  through,  the 
«utter  wheel  is  raised  by  means  of  a  small  crane,  and  the  chains,  Fig.  2, 
ud  by  the  peculiar  arrangement  of  the  nut  /  in  the  socket  vi,  the  cutter 
wheel  can  he  drawn  up  the  cylinder  without  turning  the  screw  p,  as  it 
leaves  the  nut  behind,  which  is  afterwards 
Weight  to  raise  but  that  of  the  nut.  The  < 
aew  or  finishing  cut,  after  which  the  cylindi 
'tme.  The  position  occupied  by  the  c 
and  the  bar  lifted  in  and  out  with 


wed  up,  there  being  no  other 
rs  are  tht-n  set  afresh  to  the 
may  be  connidered  perfectly 
enables  the  cylinder  to  be  placed, 
perfect  case ;    while  the  space 


occupied  by  this  machine  is  very  small  compared  with  those  where  the 
work  is  performed  horizontally;  it  is,  however,  important  that  the  base 
plate  fi  should  be  well  secured  to  the  foundations  by  strong  bolls. 

The  speed  of  this  machine  may  very  easily  be  varied,  by  having  different 
sized  riggers  or  pulleys  on  the  driving  shaft  which  conveys  the  motion  to  the 
riggers  kk.  Figs.  1  and  4. 

One  of  these  machines  may  be  seen  in  the  erecting  shops  at  her  Majesty's 
Dock  Yard,  Woolwich. 

PLATE  XLIII. 
Gbeat  Boblno  Machinb,  Bif  Messhs.  Nasmvth,  Gaskei.l,  anu  Co, 

The  mafliine  ropreaenled  by  this  Plate  is,  with  few  oxcopti<)ns,  the  naiua 
I  ibiU  last  described,  where  the  cylinder  to  be  bored  is  placed  in  a  varti- 


452  EXPLANATION  OF  THE  PLATES. 

cal  position,  wbereby  numerous  advantages  are  derived,  as  Already  ex- 
plained. 

The  motion  is  communicated  by  tbc  driving  pulley  c  to  a  bevel  innioa 
working  the  bevel  wheel  d;  the  shall  on  which  this  wheel  is  fixed,  has  on 
its  opposite  end  a  worm  for  communicating  the  motion  through  the  worm 
wheel  to  the  upright  shaft  f  and  boring  bar  a,  having  on  its  circmnferenoe 
the  grooves  a'  in  which  the  cutter  head  is  moveable,  sliding  up  and  down  ac- 
cording to  the  progress  of  the  work  ;  X-  is  a  tool  carrier  fixed  to  the  cutter  head. 
The  foundation  plate  h  forms  a  bearing  for  the  upright  shaft,  the  lower  end 
of  which  rests  in  the  step  g^  while  the  cylinder  /  is  secured  by  the  damps 
jj  to  the  supports  i  i  fixed  to  the  foundation  plate.  These  parts  are  in 
every  respect  similar  to  the  boring  machine  shewn  by  Plate  XLII.,  hy 
which  they  are  more  fully  described. 

Two  strong  piers  of  masonry  m'  support  the  entablature  m,  (for  carrying 
the  self-acting  apparatus  for  raising  and  lowering  the  cutter  head  A,)  to 
which  it  is  bolted  by  strong  holding-down  bolts  mf\  This  apparatus  consists 
of  a  rack  n  worked  by  a  pinion,  the  motion  being  transmitted  from  a  tml- 
lion  wheel  through  two  s])ur  wheels  and  pinions  o ;  the  whole  of  this  upper 
machinery  revolves  ^ith  the  boring  bar,  with  the  exception  of  tho  intenial 
wheel  or  screwed  hoop  /?,  the  consequence  of  which  is  the  small  trullion 
wheel  is  made  to  turn  on  its  axis  by  the  tread  of  the  wheel  p  in  which  it 
works,  and  thereby  ultimately  raises  the  cutter  head  6,  the  two  side 
slings  connecting  it  to  the  upper  frame  /,  to  which  is  fixed  the  rack  n. 

Tins  machine  is  of  the  largest  dimensions,  and  was  made  for  the  purpose 
of  boring  tho  large  cylinders,  10  feet  in  diameter,  for  the  Great  Western 
Steam  Navigation  Company's  vessel  the  Mammoth,  now  in  progress  at 
their  works  at  Bristol. 

PLATE  XLIV. 

Vertical  Boring  Machine,  by  Messrs.  Benjamin  Hick  and  Son, 

Bolton. 

By  this  combination  of  three  distinct  machines,  the  following  different 
operations  may  be  performed,  viz. :  boring,  drilling,  and  iace-grinding ;  it 
is  so  contrived  that  the  entablature  h^  supported  by  the  four  columns  aaao, 
carries  the  u])per  parts  of  the  three  different  machines,  consisting  of  the 
requisite  driving  machinery  for  communicating  to  them  their  respectiTe 
motions. 

That  in  the  centre.  A,  is  a  vertical  boring  machine  for  boring  cylin- 
ders of  large  diameters,  which  are  fixed  in  the  usual  way  on  the  six 
moveable  supports  h  by  the  clamps  t;  in  addition  to  which  it  is  rendered 


EXPLANATION  OF  THE  PLATES.  453 

perfectly  ste&dy  by  the  circular  frame  or  ring  /,  sliding  up  and  down  in 
grooves  on  the  back  of  the  two  middle  columns,  the  adjusting  screws  y 
being  tightened  when  the  cylinder  is  properly  placed  under  the  centre  of 
the  boring  bar  Cy  which  receives  its  motion  from  the  leather  strap  and  pul- 
leys e/,  whence  it  is  conveyed  through  the  bevel  pinion  and  wheel  e  on  the 
upright  sh&hfy  upon  which  is  also  keyed  the  spur  pinion  for  driving  the 
wheel  ff  fixed  to  the  lower  part  of  the  boring  bar  c,  and  working  in  the 
step  m ;  the  rack  and  wheel  k  gives  the  cutter  head  the  requisite  feed 
while  boring  out  the  cylinder.  The  six  supports  h  are  made  to  slide  in 
grooves  on  the  foundation  plate  n,  according  to  the  different  diameters  of 
the  cylinders  being  operated  upon ;  these,  when  properly  placed,  are  bolted 
to  the  plate  n. 

The  second  machine,  B,  is  a  vertical  drilling  and  boring  machine  for  work 
of  smaller  dimensions  than  the  machine  A.  It  is  shewn  in  the  drawing 
boring  out  the  centre  of  a  crank  x,  fixed  to  the  travelling  table  /»,  and 
slides  on  V's  on  the  frame  t/y  which  has  also  a  motion  at  right  angles,  on 
the  bed  o  fixed  to  the  foundation  plate.  The  drilling  bar  1/  is  lowered  by 
the  screw  u^^  according  to  the  feed,  its  motion  being  conveyed  to  it  from 
the  pulley  q  and  strap  /  to  a  spur  wheel  and  pinion  not  shewn  in  the 
drawing,  the  pinion  is  on  the  same  spindle  as  the  pulley,  and  the  wheel  on 
that  of  the  reversed  cone  tt,  by  which  it  is  carried  to  the  square-threaded 
screw  u''  by  two  pairs  of  small  bevel  wheels  »,  fixed  on  an  horizontal 
spindle  and  working  in  suitable  bearings  on  the  carriages  u'^'  v^'\  The 
apparatus  for  raising  the  drilling  bar  is  the  same  as  that  described  by  Plate 
XXXII.,  which  consists  simply  of  two  small  chains  fixed  to  the  bar  t/ 
and  working  round  the  pulleys,  its  opposite  end  being  attached  to  a 
weight  t?. 

The  third  machine,  C,  is  for  the  purpose  of  grinding  up  the  fiEu:es  of  rings 
for  metallic  pistons,  conical  valves,  &c. ;  the  travelling  table  p  of  this  ma- 
chine is  in  every  respect  similar  to  that  of  B,  and  upon  it  is  placed  the 
piston  ^  to  be  ground ;  the  upright  rod,  receiving  its  motion  from  the  pulley 
and  strap  /,  is  kept  in  a  vertical  position  by  the  cross  frame  «,  while  the 
shaft  r  and  grinding  plate  are  connected  to  the  lower  end  of  the  rod, 
and  are  occasionally  raised  for  examining  the  surface  being  ground. 

The  different  motions  given  to  these  machines  are  quite  independent  the 
one  of  the  other,  by  which  means  any  one  of  them  can  be  worked  sepa- 
rately. The  whole  is.  placed  on  a  suitable  strong  foundation  of  stone. 
After  the  large  cylinder  is  bored,  it  is  raised  from  its  position  by  a  crane 
placed  on  the  floor  above. 


4f54f  EXPLANATION   OF   THE   PLATES. 

PLATE  XLV. 
Machine  for  planing  Iron,  by  Nicholas  Forq,  Clockmakxb. 

This  Plate  represents  an  elevation  of  a  planing  machine,  which  was  in- 
vented as  early  as  the  year  1751,  for  the  purpose  of  planing  the  pump 
harrels  used  at  the  Marley  Water  Works  near  Paris.  These  pumps  con- 
sisted of  i;«Tought  iron  segments  bound  together  by  strong  hoops  or  strspi; 
for  tliis  purpose  M.  Forq  erected  this  machine  near  Mauberge  in  Fiance. 

It  consisted  of  a  small  spur  wheel  e  placed  between  two  trundle  wheeb 
A  and  t  in  the  same  vertical  plane.  The  lower  trundle  t  was  put  in  motion  bj 
manual  labour  being  applied  to  the  ninch  handle  p ;  on  the  same  spindle 
was  fixed  a  fly  wheel  c  to  regulate  the  motion,  while  the  upper  tnmdle 
wheel  h  communicated  it  to  a  laige  spur  wheel  b  above,  one  half  of  its  dr- 
cumference  only  being  funiished  with  teetli.  On  the  axis  of  this  wheel  h 
a  wooden  pulley  a  of  a  very  large  diameter  wbb  fixed,  over  its  periphery 
an  endless  bond  or  cord/"  was  made  to  pass,  and  also  over  the  im»t1|gr  po]. 
leys  klmn  op^  to  which  was  fixed  the  cutter  or  tool  fromc.  This  labot  or 
cutter  was  then  drawn  backwards  and  forwards,  between  two  laige  panllel 
bars  of  square  iron  placed  horizontally,  for  giving  tlie  alternate  motion  to 
the  tool,  which  formed  a  kind  of  cross,  the  cutting  surface  being  suited  to 
the  circular  figure  of  the  interior  of  the  barrel,  and  bored  or  planed  it 
regularly. 

The  bearings  of  the  «j)in(llcs  were  fixed  on  cross  timbers,  in  a  similar 
wav  to  the  fnimc  for  currvinir  tlie  work  boin^r  planed. 

The  segments  forming  this  barrel  hud  been  previously  planed  separately, 
by  a  similar  process  to  that  already  described ;  its  parts  having  been  well 
forged  to  the  required  sha])e  j)revious  to  their  admittance  in  the  planing 
machine,  when  all  the  joints  were  made  perfect. 

By  this  machine  M.  Forcj  seems  to  have  prepared  all  the  barrels  he  re- 
quired, as  he  says,  with  the  greatest  accuracy,  which  varied  in  their 
diameters  from  10  inches  to  4  feet,  and  from  7  feet  to  10  feet  long. 
The  inventor  asserts  that  he  had  seen  nine  barrels  7  feet  long,  of  which 
eight  had  a  diameter  of  10  inches,  and  the  remaining  one  15  inches,  filled 
with  water  for  three  months,  and  were  perfectly  water-tight. 

A  pump  barrel,  10  inches  diameter  and  7  feet  long,  consisted  of  nine 
staves  or  segments,  held  together  by  1 2  uTought  iron  hoops  three  inches 
wide,  the  extremities  terminating  with  collars  two  inches  broad.  The 
hoops  were  half  an  inch  thickness  of  metal. 

This  apparatus  is  evidently  incomplete  for  intuit  of  the  attachment  of  the 
planing  tool,  and  the  reciprocatory  motion,  which  is  not  shewn  in  the  original 
drawing. 


EXPLAKATION    OF    THE    PLATES. 


PLATE  XL  VI.— Flo.  1. 


Mlllwhioiit's  Planino  Machtkb,  dy  Messrs.  Nasmvi 

AND   Co. 


Tliis  Pittlc  reprpscnta  two  iliffcrent  modes  of  planing.  Pig,  1  shews  the 
tool  moveable;  Fig.  2,  the  tool  fixeil.  By  the  first  arrangement  many  od- 
vantagcs  arc  derived,  such  as  planing  the  parts  of  very  large  and  heavy  ob- 
jects, whose  great  weight  and  size  would  otherwise  render  them  quite 
inadmiseible  id  the  machines  of  ordinary  construction,  where  the  work  has 
to  be  fixed  on  a  sliding  bed  and  moved  along  the  machine  with  it.  It  is 
evident  by  this  contrivance  tliat  a  very  great  saving  of  power  is  effected, 
since  ihe  only  moving  part  of  this  machine  consists  of  a  small  traversing 
fnune  for  carrying  the  slide  and  tool. 

By  referring  to  Fig.  I,  it  nill  be  seen  that  tlijs  machine  rests  in  a  slip  or 
pit  mode  for  the  purpose,  with  steps  to  descend  into  it.  The  two  side 
f^Bmes  a  are  bolted  to  the  ground,  and  stretching  bolts  connect  them  above. 
There  are  also  two  cross  carriages  for  carrying  the  shaft  b,  and  a  bed  plate 
hftring  a  series  of  apertures  through  which  any  piece  of  work  can  be 
wcureO,  they  can  be  raised  or  lowered  to  any  height  by  four  side  screws. 
The  ftnme  e  for  carrying  the  slide  ivorks  liockwards  and  forwards  along  the 
edges  of  tlie  bed  a,  and  by  nicaus  of  a  screw  and  handle  rf,  its  position  can 
easily  be  altered. 

It  is  evident  from  the  above  description,  that  in  eases,  such  as  planing  the 
on  large  heavy  shufbs,  lliis  machine  is  found  to  be  very  useful 
ml  economical,  for  the  surface  to  be  planed  is  but  very  small  com- 
d  with  the  lougth  and  bulk  of  the  work  which  otherwise  would  be 
veiling  along  tlie  bed,  occa-siuning  both  waetu  of  time  tmd  power.  The 
ingth  given  to  the  cut  can  be  altered  at  pleasure. 


PLATE  XLVI.— Fio.  2. 


Plahino  Machine,  by  Messrs.  Nasuvth,  Gaskell,  i 


>  Co. 


Tlie  |>erspective  drawing  shewn  by  Fig.  2,  represents  a  planing  machine 
iF  ordinary  constniction,  ivherc  the  tool  is  fixed  and  the  work  moveable. 
It  consists  simply  of  a  bed  a  uhoiit  nine  feet  three  inches  long,  upon  which 
e  iniyelliiig  Inbh'  b  works  backwards  and  forwards.  The  wheels  for  work- 
ing this  table  arc  so  arranged  ae  to  bring  it  bock  after  the  work  ha«  been 
It  a  niueh  greater  speed,  thereby  saving  time  to  a  very  great  extont. 
e  two  side  frames  ee  have  on  their  faces  the  upright  slots  or  groovds  ifif, 
D  wliicli  the  cross  frame  </  is  raised  and  lowered  at  pleasures  to  reguUto  ita 


456  EXPLANATION  OF  THE  PLATES^ 

height  according  to  the  magnitude  of  the  work  being  planed.  TUa  is 
easily  performed  by  the  handle  wheel  and  large  screw  k  woildng  in  a 
nut  on  its  back.  The  sliding  frame  e  for  carrying  the  tool  ^  is  then 
brought  to  its  proper  situation  by  means  of  the  screw  and  handle/^  after 
which  the  machine  is  set  to  work  by  running  the  leather  strap  from  a  looae 
pulley  or  rigger  to  one  fixed  on  the  shafl. 

The  table  b  has  the  usual  mortices,  by  wliich  the  work  is  clamped  to  it 
by  strong  bolts. 

PLATE  XLVn. 
Planing  Machine,  by  Messrs.  Nasmyth,  Gaskell,  and  Co. 

The  only  difference  between  the  principle  upon  which  this  machine  wmksy 
and  that  last  described  by  Fig.  2,  Plate  XL  VI.,  is  in  the  motion  by  which  the 
sliding  table  is  brought  back  after  it  has  travelled  the  length  of  ihe  bed. 
From  its  great  similarity,  it  is  termed  the  mangle  motion,  and  by  the  great 
spcc<l  obtained,  is  found  very  useful. 

Tlie  bed  a  rests  on  six  columns,  connected  together  by  diagonal  inimet 
to  increase  its  stability ;  and  the  sliding  tabic  b  running  back\i'ards  and  fbr- 
^urds  on  this  bed,  has  the  usual  mortices  in  it.  The  upright  frames  H 
are  fixed  to  the  bed  by  bolts ;  in  the  two  grooves  of  this  frame,  the  cross 
frame y  is  made  to  slide  up  and  down  to  suit  the  work  to  be  planed,  by  means 
of  the  long  screw  and  handle  /,  after  which  it  is  securely  fixed  to  them. 
The  sliding  frame  for  carrying  the  tool  can  be  placed  in  any  situation  along 
the  frame y,  by  turning  the  handle  and  screw  //,  by  which  it  may  be  brought 
in  close  contact  with  the  work ;  it  can  also  be  placed  at  an  angle  by  loosen- 
ing the  screws  in  the  short  mortices  shewn  on  its  face.  Tlie  tool  can  be 
taken  out  and  replaced  by  another,  by  two  small  adjusting  screws,  by 
which  it  is  fixed. 

The  motion  is  communicated  to  the  machine  by  a  leather  strap  working 
on  the  riggers  c.  The  spindle  for  carrying  them  being  supported  on  the  top 
of  a  small  column  by  its  side ;  on  the  opposite  end  of  the  spindle  is  the 
small  pinion  working  alternately  inside  and  outside  of  the  tnillions  on  the 
large  wheel  </,  the  distance  travelled  by  the  table  being  regulated  by  a 
stop  on  its  periphery ;  this  is  effected  by  the  spindle  sliding  backwards  and 
forwards  in  the  groove  ^,  situated  on  the  side  of  the  machine.  On  the 
shaft  of  the  trundle  wheel,  is  a  large  wheel,  round  which  an  endless  chain 
is  placed,  ])assing  over  two  small  pulleys  y^,  which  pulleys  arc  fixed  to  the 
frame  by  a  screw  ;  one  of  these  only  can  be  seen,  the  other  being  on  the 
opposite  end  of  die  machine ;  the  chain  is  fixed  to  both  ends  of  the  bed, 
and  communicates  the  alternate  motion  to  the  sliding  table ;  dose  to  the 


EXPLANATION    OF   THE    PLATES.  457 

gioove  bearing  e,  (which  forms  part  of  the  diagonal  frame,)  is  a  Bmall  rod  ff 
connected  to  a  spindle  by  a  series  of  levers  and  bell  crunks,  by  which  the 
Ujinght  Toii  IB  made  to  rise  and  fall  nllernatcly,  n'orkiiig  the  ratchet  wheel 
fixed  to  the  screw  A,  therehv  giving  a  self-acting  motion  to  the  cutter  aJong 
the  frame  _/.      This  niHchino  is  capable  of  planing  work  about   8  feet  long, 

I  by  3  feet  6  inches  wide.     The  length  being  regulated  by  the  distanco 
Invclled  by  the  sliding  tabic. 
The  frame  for  carrying  all  the  machinery  is  well  bolted  to  tie  floor. 


PLATE  XLVIl.  A. 
Planing  MAcamB,  by  Mbssrs.  Nasmyth,  Gaskbll,  and  Co. 


^mie 
^lobe 


This  description  of  planing  machine  is  the  some  as  that  represented  by 
Kg.  2,  Plate  XLVI.,  where  the  work  to  he  planed  is  filed  to  n  traTelling 
table  d,  moveable  along  lie  surface  of  a  bed  a,  and  made  to  travol  both 
backwards  and  forwards  by  a  self-acting  apparatus,  which  is  also  connected 
with  tlie  tool  o,  the  motion  being  in  the  first  instance  communicated  to 
Mther  one  or  the  other  of  the  two  pulleys  e  or/;  that  marked  e  is  fixed  on 
a  shaft  y*,  and  conveys  the  motion  through  the  wheels  and  pinions  c"  to  a 
n  working  the  rack  h,  screwed  to  the  under  side  of  the  table  d.     This 

riBTelling  table  has  a  number  of  mortices,  by  means  of  which  the  machinery 
to  be  operated  upon  is  clamped,  while  the  bed  a  on  which  it  slides  has  one 
a  V  form,  and  the  other  a  flat  smooth  surface,  Fig.  2.     The 
two  upright  side-frames  i/b  are  Bccnrcd  to  the  bed  n,  and  are  connected  to- 
gether at  the  top  by  a  cross  frame  c ;  these  two  frames  have  long  vertical 
mortices  in  which  the  sliJingcrosaframe/ismovenhleaccording  to  the  dimen- 
sions of  the  work  placed  on  the  table,  and  can  be  raised  or  lowered  by  the 
wheel  nnd  square- threaded  screw  /' ;  a  second  frame  m  is  also  moveable  along 
the  cross  frame  /  by  the  ratchet  wheel,  square- threoded  screw,  and  wheel  m', 
suitable  to  any  required  position,  while  a  third  motion  is  obtained  by  the  slide 
1^^  n  (for  carrjing  the  tool «)  being  moveable  on  the  frame  m  by  the  wheel  and 
^^Mcrew  «'.      By  the  following  arrangement  the  reversing  motion  of  the  table 
^^knd  that  of  the  frame  m  and  tool  are  made  self-acting.  On  one  side  of  the  tra- 
I^Kvelliug  table  are  two  moveable  studs _/_;,  (regulated  according  to  the  distance 
ihe  table  is  intended  to  travel,)  which  coming  in  contact  with  the  lever  i" 
strike  against  the  upper  part  /,  and  communicate  the  motion  on  one  side 

Erod  and  lever  k,  and  also  to  the  upright  rod  i,  to  which  is  fixed  the 
ible  stud  and  pauI  i^  for  working  the  ratchet  wheel  m'  and  screw,  oa 
y  described,  the  counterhaionue  weight  k  causing  tlie  paul  to  rise  for 
'wed  stroke.  Two  guide  pieces  ore  screwed  to  the  frame  i  for  the 
it  rod  t  to  work  in.     On  the  opposite  side  of  the  lever  i"  is  n  short 


458  EXPLANATION  OF  THE  PLATES. 

connecting  rod  (for  reversing  tLe  motion  of  the  travelling  table)  fised  to 
the  bell  crank  working  the  rods  shewn  by  the  dotted  lino  in  Fig.  S  under 
the  table,  which  ultimately  transmit  the  motion  to  the  disengaging  ^ipa- 
ratus,  consisting  of  an  upright  lever,  counterbalance  weight,  and  cross  bar  J', 
with  two  short  projecting  bars  for  the  leather  strap  to  work  between,  and 
by  which  it  is  passed  from  the  pulley  e  to  that  of/^  whereby  the  motion  if 
immediately  reversed,  the  spur  wheel  and  pinion  f  causing  the  pinion  to 
work  the  rack  ^  in  a  contrary  direction,  there  being  three  wheels  in  one 
instance  and  two  in  the  other.  Both  the  pulley  f  and  pinion  f  are  fixed 
to  a  cylinder  working  freely  on  the  spindle  /. 

The  travelling  table  is  generally  made  to  return  at  a  much  greater  velo- 
city than  that  required  while  the  tool  is  acting,  by  which  means  consider- 
able time  is  economized. 

When  the  machine  is  at  rest,  the  leather  strap  runs  on  the  loose  pnlley  g 
without  producing  any  effect  on  the  working  parts  of  the  machine,  in  which 
case  the  disengaging  bar  i^  is  situate  in  the  position  shewn  by  the  pkn 
Fig.  3. 

A  frame  p  and  standard  q  for  suppordng  the  shafts  /  and  the  seTenl 
parts  of  tlie  machine  is  fixed  to  the  side  of  the  bed  a,  and  the  tool  o  can 
be  changed  at  pleasure  by  unscrcT^ing  the  screws  which  hold  it  in  its  place. 


PLATES  XLVII.  B.  AND  XL VII.  c. 

Planing  Machine,  driven  by  Steel  Belts,  by  Benjamin  Hick 

AND  Son,  Bolton. 

These  two  Plates  represent  a  planing  machine  of  a  different  description 
from  those  generally  used,  and  of  unusual  size,  and  o^\ing  to  the  peculiar 
motion  given  to  the  tool,  it  cuts  both  ways,  on  Whitworth's  patent  plan.  It 
is  shcu-n  in  Figs.  1  and  2,  planing  the  faces  of  the  ports  of  a  cylinder  for 
a  steam  engine. 

The  arrangements  of  its  parts  consists  of  two  upright  side  framea  a  a, 
fixed  below  the  surface  of  the  ground,  on  which  are  placed  the  cross  and 
longitudinal  bearers  b  /;,  for  carrying  the  work  to  be  planed.  On  the  top  of 
the  frames  a  a  are  bolted  two  long  side  frames  dd  supported  also  by  the 
three  standards  eee;  the  sliape  of  these  two  frames  dd^Bs  will  be  seen  by 
Fig.  2,  is  of  the  V  form,  and  upon  them  the  cross  table  c  for  carrying  the 
tooiy  works ;  there  are  two  bearings  on  this  table  for  supporting  the  hoUow 
cylindrical  bur,  upon  wliich  the  frame  i'  is  moveable,  (by  meana  of  a 
passing  through  the  hollow  bar,)  according  to  the  position  of  the  too]  j 
the  face  of  the  work  being  planed ;  a  long  key  is  provided  on  the 


EXPtANATlON  OF  THE  PLATES.  459 

Aneai  bar,  to  prcTent  tlie  carriage  i"  from  altering  its  pontion  by  tnming 
on  it. 

On  the  front  port  of  the  frame  i"  is  a  circular  bos,  which  moves  at  each 
end  of  tlie  cut ;  in  it  the  small  cyliniler  for  carrying  the  tool  revolves ;  tliis 
front  part  of  the  frame  i'  can  Lc  placed  at  any  angle,  by  means  of  grooves 
on  its  facen,  two  small  bolts  and  iiiita  on  either  side  fixing  it,  when  pro- 
perly adjusted,  while  the  tool  is  kept  in  contact  with  the  fiu»  of  the  work 
by  the  handle  and  screw  above  it. 

The  working  of  this  machine  is  as  follows :  the  motion  being  commnni- 
cated  to  the  pnllevB  by  the  leather  straps  //,  the  one  for  working  the  ma- 
chine forward,  and  the  other  for  working  it  in  a  contrary  direction,  convoy 
it  through  the  two  spur  wheels  and  pinions  ffff,  to  the  puUeye  fi,  fixed  on 
two  strong  shatls  extending  across  the  extreme  ends  of  the  machine  under 
the  floor,  one  of  these  pulleys  being  situate  at  each  comer  of  tLe  machine, 
uid  round  them  the  steel  belts  work,  the  ends  of  which  are  attached  to  the 
projecting  levers  m  on  the  sides  of  the  long  cross  bar  i,  and  secured  by 
tightening  screws;  the  operation  of  reversing  con  be  effected  at  any  point 
by  levers  and  a  rod,  extending  the  whole  length  of  the  bed,  and  Iios  a 
moveable  stop  for  that  purpose. 

The  friction  of  tlieso  belts  upon  the  pulleys,  drives  the  machine,  and  is 
snfficient  for  the  heaviest  cats,  its  only  moveable  parts  being  the  carriage  c 
and  its  appendages,  while  the  object  under  operation  of  planing  is  a  fixture, 
by  which  means  the  same  amount  of  power  is  necessarily  required  for  eitlier 
a  heovy  or  a  light  casting ;  the  surface  it  is  capable  of  planing  being  30  feet 
long,  and  9  feet  ft  inches  wide ;  when  large  surfaces  such  us  lathe  beds  are 
to  be  planed,  the  plolform  b  is  raised  to  a  level  with  the  floor. 

^ole. — A  similar  tool  was  also  contrived,  some  years  ago,  by  the  editor  of 
thU  work. 

^^  The  pocnlinrity  of  this  machine  is  its  adaptation  to  two  different  pnr- 
poBes,  so  closoly  connected  to  one  another,  tliat  in  the  construction  of  boilers 
of  any  description,  or  in  fact  to  any  work,  where  wrought  iron  plates  are 
used,  either  to  be  cut  or  punched,  it  is  found  very  useful ;  by  the  arrange- 
ment of  the  parts  of  this  machine,  tlils  is  effected  at  the  same  time  and  by 
the  same  motion. 

tThe  large  cast  iron  frame  or  standard  d,  carries  the  shaft  r,  on  one  end 
wbioh  is  fixed  an  eccentric,  Pig.  I ,  for  nuong  and  lon-ering,  or  giring  tho 


PLATE  XLVIIl. 


PUKCBINO  AND  PLAIE-CUTTINO  MaCRINB,  BY  MBaSKS.  NASUVTH, 

Gaskbll,  and  Co. 


460  EXPLANATION  OF  THE  PLATES. 

alternate  motion  to  the  slide /*;  on  the  other  extremity  of  this  aame  ihaft, 
is  the  large  spur  wheel  c,  put  in  motion  bj  the  pinion  on  the  driving  shaft 
on  which  is  hung  the  fly  wheel  by  for  steadying  the  motion^  and  also  the 
two  pulleys  or  riggers  aa^  the  one  keyed  to  it,  while  the  other  is  aDowed  to 
run  loose,  when  the  machine  is  not  working.  The  outer  carriage  jp  supports 
the  hearing  of  this  latter  shaft. 

On  the  upper  part  of  the  frame  d  is  the  steel  cntter  ^,  secured  to  it ;  a 
similar  one  is  placed  in  the  reverse  position  on  the  top  of  the  sliding  frsme 
fy  die  lower  part  having  fixed  to  it  the  punches  h  for  punching  the  plateSi 
which  work  freely  in  the  dies  i  screwed  to  the  frame  d;  there  is  a  stop  J 
on  the  under  side  of  the  jaws  of  the  frame,  for  preventing  the  plate  from 
rising,  as  it  has  a  tendency  to  rcmun  fixed  to  the  punches,  which  it  is  ohiiged 
to  quit,  when  it  comes  in  contact  with  the  stop. 

From  the  above  description,  the  mode  of  operation  will  be  easily  under- 
stood. The  motion  being  given  to  the  fixed  pulley  from  the  driving  shaft  of 
the  building,  is  then  communicated  to  die  slide  by  the  pinion  and  wheel, 
causing  the  eccentric  shaffc  e  to  revolve,  thereby  giving  the  up  and  down 
motion  to  the  punches,  and  also  the  cutters  ffff  alternately,  which  in  the  side 
elevation.  Fig.  2,  is  shei^-n  cutting  a  plate.  The  slide y  works  in  a  V  slide 
fixed  by  six  screws,  die  steel  cutter  being  also  fixed  by  three  of  a  similar  siae. 

The  punching  operation  performed  by  this  machine  is  made  self-actiiig^ 
by  the  following  arrangement.  The  plate  to  bo  punched  is  secured  in  the 
usual  way  by  clamps  to  the  travelling  tabic  ky  its  four  wheels  running  on  a 
railway  of  triangular  bars,  on  the  bed  plate ;  two  carriages  o  are  fixed  to  the 
under  side  of  the  travelling  table,  for  supporting  the  notched  bar  n  bolted  to 
them. 

As  the  large  spur  wheel  c  revolves,  the  pin  7w  comes  in  contact  with  the 
lever  connected  to  the  rod  /,  by  which  tlie  bar  n^  and  consequently  the 
table,  is  advanced.  This  done,  the  table  is  again  drawn  back  to  its  former 
position,  by  the  chain  shewn  in  Fig.  1 ,  when  the  operation  is  repeated. 

Messrs.  Nasmyth,  Gaskell,  and  Co.  have  made  subsequent  improvements 
in  this  machine,  by  altering  the  position  of  the  apparatus  for  cutting  the 
plates  of  boilers,  to  the  opposite  side  of  the  punching  apparatus,  while  the 
frame  work  is  so  adapted  as  to  carry  the  driving  parts  in  the  centre,  the 
same  shaft  working  both  machines. 

PLATE  XLIX.— Fios.  1  and  2. 

Plate-bending  Machine,  by  Messrs.  Fairbairn  and  Co. 

The  variety  of  forms  given  to  different  boilers,  according  to  the  natUK 
of  the  steam  for  wluch  they  are  intended,  renders  the  machine  shewn  by 


EXPLANATION    or    THE    I'l.ATF.S.  i-G] 

I  and  2  of  this  Plate  quito  indispensable  to  b  boiler  manufaotarer-  It 
will  easily  be  «een  by  ekantining  these  drawings,  buiv  a  t^at  wrought  iron 
pUlu  may  l>e  brought  to  any  required  curve,  by  passing  it  between  the 
rollers  A  6,  which  are  regulated  by  the  large  adjusting  screws  ^y. 

The  driving  and  loose  riggers  a  a  have  the  motion  comiuunicated  to  them 
"hy  a  leatlier  strap,  from  a  shaft  worked  by  the  engine  connected  to  the  esta- 
blishment, the  spur  wheel  e  on  an  intenaediate  shaft,  then  conveys  it  by  a 
n  to  the  large  wheel,  on  which  is  fixed  one  of  the  rollers  b,  the  other 
being  worked  by  the  two  pinions  on  the  opposite  side  of  the  machine.  The 
fiy  wlioel  (1  is  placed  on  the  driving  or  rigger  shaft,  and  by  its  great  weight 
^vee  a.  regular  steady  motion  to  the  different  working  [>arts  of  the  machine. 
The  roller  bearings  aje  adjusted  by  the  large  square-tlireaded  screws  working 
in  the  two  side  frames  e  for  supporting  them.  Much  additional  strengtli  is 
^ven  to  this  machine,  by  the  stretching  bolts//,  which  bind  it  together. 

By  placing  a  handle  on  one  of  the  arms  of  the  fly  wheeltl,  this  machine 
inight  be  put  in  motion  by  manual  power. 

PLATE  XLIX.— Pigs.  3  and  4. 

Vice  fob  ccttino  Boilkr  or  othbb  WBonoHi  Ibon  Plates,  by  Messrs. 
Nasuyth,  Gaskrll,  and  Co.,  Manchester. 

Figa.  3  and  4  aliew  a  simple  contrivance,  by  which  n-rought  iron  plate 
■n  be  held  secure  in  a  frame,  while  the  edges  are  cut  with  a  chisel.  The 
Trnme  conaals  of  the  upper  and  lower  parts  of  a  vice  a  and  c,  the  fates  of 
which  are  hardened  steel,  iwtween  these  a  wrought  iron  plate  d  is  placed, 
where  it  is  securely  clamped  by  tightening  the  two  nuts  on  the  large  squarc- 
btreaded  screws  ft,  keyed  through  the  frame,  which  also  connect  the  ma- 
diine  to  the  foundation  of  the  building. 

The  chisel  t  for  cutting  this  plate  is  shewn  by  Fig.  4,  by  which  the  work 
IB  performed  in  a  very  perfect  way. 

The  lower  vice  frame  « is  much  strengthened  by  tlie  ribs  shewn  by  Fig.  4, 
which  give  it  a  Krm  bearing  on  the  ground. 

PLATE  L.— Fios.  I  and  2. 

PuNCHiNd  Machine,  by  Mhssrs.  Kinmond,  HuTTrtN,  and  Stbrl, 

UUNDRE. 

These  two  figures  represent  an  elevation  and  an  end  view  of  a  machine 
T  punching  holes  in  wrought  iron  plates  for  boilers  and  other  purposes. 
The  Rtrong  frame  J,  which  carries  the  whole  nf  the  machiner)-,  is  maile 


462  EXPLANATION  OF  THE  PLATES. 


of  cast  iron,  and  is  Very  firmly  fixed  to  the  floor  of  the  bnildiiig.  The  pat 
leys  a  receive  the  motion  from  a  line  of  shafts ;  one  of  these  runs  looM^ 
and  tlic  8tra])  is  thrown  on  it  when  the  machine  is  at  rest*  The  qpeed 
given  to  tlie  pimching  spindle  £/  is  reduced  hy  two  pairs  of  spur  vdiedb  wdA 
pinions  c  c ;  upon  the  end  of  the  spindle^  is  fixed  the  eccentric  or  Gamy  ivindi, 
as  it  revolves,  raises  and  lowers  the  slide  h  for  carrying  the  punch  e,  'wUdi 
works  in  a  die  y^  large  enough  in  diameter  to  fit  it,  and  through  thia  die 
passes  the  circle  of  the  plate  cut  out  hy  the  punch. 

A  very  uniform  motion  is  obtained  by  phicing  the  heavy  fly  wheel  i  oe 
the  main  driving  shaft. 

The  different  ports  of  this  machine  can,  without  difficulty,  be  taken  eat 
and  replaced  by  otlicr  suitable  to  larger  or  smaller  work. 

PLATE  L.— Fios.  3  and  4. 
Messrs.  B.  Hick  and  Son's  Mandbil  for  rxpandino  Kxnqs. 

The  mandril  in  a  lathe,  is  that  part  upon  which  tlic  work  to  be  turned  k 
placed  or  fixed,  consequently  different  mandrils  arc  required  to  suit  the  va- 
rious kind  or  forms  of  macliincry  to  be  turned ;  by  tliis  contrivance  of  Mr. 
HickK,  rings  of  very  different  diameters  may  be  turned ;  a  few  words  will 
give  an  idea  of  the  mode  in  which  this  is  performed.  The  spindle  or  man- 
dril a  is  of  a  cylindrical  form,  having  a  square-threaded  screw  at  one  end  ; 
on  tlie  larger  or  middle  part  of  this  mandril  arc  four  grooves,  in  which  the 
conical  pieces  care  made  to  slide,  and  have  on  their  circumference  the  ring  d 
to  be  tunied  ;  a  tightening  cone  ^  of  a  cylindrical  shape  is  then  placed  on 
the  screw,  and  by  screwing  the  nut,  presses  it  against  Uie  four  pieces  c,  thos 
expanding  them  till  the  ring  becomes  quite  securely  fixed  on  their  circum- 
ference. 

By  this  means,  rings  of  various  diameters  may  be  turned,  within  the  limits 
intended  by  the  mandril. 

PLATES  LI.  and  LIL 

Machinr  for  Punching  Boiler  Plates,  by  Messrs.  Maudslat,  Sons, 

AND  Field. 

Fig.  1  is  an  elevation,  Fig.  2  a  plan,  and  Fig.  3  a  side  view  of  a  punch- 
ing machine  on  a  very  improved  principle,  whereby  the  plates  required  for 
boilers  and  other  purposes  may  be  punched  T^ith  the  greatest  possible  accu- 
racy, insuring  at  the  some  time  very  superior  workmanship  and  gml  dis- 
])atch.  It  is  usual  in  all  ordinary  punching  machines,  fiist  of  all  to  nnk 
out  the  rivet  holes  in  the  plate,  by  a  template,  with  white  pamt^  end  dwn 


EXPLANATION    Of    THK    I'LATES. 


i63 


l 


to  place  it  as  nenr  as  the  eye  will  pennit  under  the  punch ;  by  the  coDtriv- 
ttace  of  this  machine,  this  operation  is  entirely  dispensed  with,  it  being  only 
Seccseary  to  fix  the  plate  to  a  travelling  table,  and  then  to  adjust  the  various 
Jiarts  of  the  machine  to  the  proper  distance  reciuired  between  the  rivets. 

The  large  cast  iron  frame /I  carries  the  several  parts  of  the  machinery, 
and  also  the  two  plummer  blocks  for  supporting  the  hearings  of  the  lying 
Aaft  a  running  the  whole  length  of  the  building,  for  the  purpose  of  work- 
ing other  machines ;  this  frame  is  securely  bolted  to  the  wall,  which,  added 
to  its  own  weight,  gives  it  great  stability. 

On  the  shafts  a,  connected  together  by  the  two  coupling  boxes,  is  placed 
the  crank  b,  the  motion  being  communicated  through  the  crank  pin  to  a 
connecdng  rod  c,  to  which  is  attached  the  upper  lever  d,  being  always  at 
work,  while  the  shaft  a  is  revolving ;  the  fulcrum  of  this  lever  is  on  the 
(tKtae  p.  A  lower  lever  e  for  raising  and  depressing  the  punching  frame  / 
has  also  its  fulcrum  on  the  same  frame ;  this  last  lever  working  only  when 
the  punching  operation  is  being  performed.  On  the  top  of  tlie  frame  p  is 
A  lever  and  long  rods  e'  for  engaging  and  disengaging  the  machinery.  The 
mde  view  represented  by  Fig.  3,  shews  the  machino  at  work,  and  by  draw- 
ing down  the  lever  e",  connected  by  the  rods  to  the  eounterbolance  weight, 
which  will  allow  it  to  remain  steady  in  any  position,  they  are  disengaged 
^ra  the  pin  on  the  lever  </,  which  at  once  cesses  to  communicate  the  mo- 
tion to  the  lower  or  punching  lever  e ;  thus  it  will  cosily  be  understood  that 
parts  of  the  machine  constantly  at  work,  are  the  lying  shaft  a,  the 
crank  b,  the  connecting  rod  c,  and  the  upper  lever  d. 

The  alidcyfor  carrying  the  punclies,  works  between  V's,  Fig.  2;  one  is 
fixed  to  the  frame  yi  by  adjusting  screws,  both  on  its  face  and  sides,  and  by 
the  curious  shape  given  to  the  end  of  the  lever  e,  tliia  slide  is  made  to  rise 
and  fall,  the  counterbcdance  weight  and  lever  p  being  connected  by  two 
short  links ;  on  the  bottom  of  the  slide  is  screwed  a  small  frame  for  carry- 
ing the  punches,  wliich  may  be  taken  out  and  replaced  by  others.  The  dies  i 
in  which  the  punches  work  are  placed  in  a  frame  bolted  to  the  frame  p,  and 
by  unscrewing  the  small  adjusting  screws,  these  niny  be  taken  out  and  re- 
placed by  others  suitable  to  the  different  sized  punches  that  may  be  required. 
On  tlie  under  side  of  the  frame  p  is  screwed  a  small  stop,  by  which  the 
circle  punched  from  the  plate  ia  forced  out  os  the  slide  rises. 

in  the  front  of  the  machine  is  placed  a  long  table,  supported  by  the  car- 
I,  consisting  of  two  columns  and  diagonal  frame,  having  bolted  to 
^em  two  long  bars,  upon  which  the  moveable  table  is  made  to  slide 
tilis  table,  as  nill  be  seen  bv  Fig.  2,  him  a  number  of  holes  by  which  thi 
jlaxe  to  ho  punched  is  secured  by  clamps ;  tl  is  odviuiced  by  the  rope  or  chain 

passing  under  the  pulley.  Figs.  1  and  3,  and  over  a  tiecund  one  hung  from 
iling  of  the  building,  to  which  is  ctiuuccted  ti  weight  sullicicntly  hi 


holes  by  which  the  ^ 

by  the  rope  or  chain  I 

icund  one  hung  from  ^fl 

;ht  sullicicntlv  heavv  ^^^M 


4G4  EXPLANATION  OF  THE  PLATES. 

to  draw  fon^'ard  the  table  and  plate  fixed  to  it ;  afler  traTeUing  the  kagdi 
of  the  table  o,  it  may  be  brought  back  by  taming  the  windi  handle  wai 
spindle  /,  on  which  is  a  pinion,  for  working  the  rack,  fixed  to  the  under  ade 
of  Uie  moveable  table  m. 

By  a  very  ingenious  contrivance,  forming  part  of  this  machine^  the  met 
holes  of  boilers  may  be  pimched  to  very  different  pitches,  a  thing  Teiy  mtA 
wanted  in  such  cases  as  the  repairing  of  old  boilers,  or  replacing  an  oU 
plate  by  a  new  one,  where  it  is  of  the  utmost  importance  to  have  As 
rivet  holes  coinciding  with  the  greatest  accuracy,  which  may  he  hetts 
understood  by  supposing  tliat  in  the  length  of  a  plate,  one,  two,  or  dnw 
additional  rivet  holes  may  be  required,  which  distance  would  have  to  hd 
equally  divided  in  the  whole  length  of  the  plate.   A  description  of  this  pvt 
of  the  machine  will  fully  shew  how  this  operation  is  performed.     On  die 
end  of  the  punching  lever  e  is  screwed  a  small  plate  Hith  a  pin/^,  adjostahle 
by  a  screw  working  in  the  short  mortice.  Fig.  3 ;  on  the  frame  p  is  holled  a 
fulcrum  piece  for  the  lever  and  counterbalance  weight y;  to  this  lever  a  long 
rod  is  connected  at  its  lower  end,  passing  under  the  table  m,  and  the  lever 
r  is  fixed  to  it,  as  will  be  seen  by  the  dotted  lines  in  Fig.  2  ;  it  has  also  two 
stops  or  projecting  pieces  /  /  on  its  surface.    A  second  lever  «  has  its  fnl- 
crum  fixed  to  the  table  o,  one  end  of  tliis  lever  being  connected  to  the  lever 
r,  while  the  opposite  end  has  on  its  surface  two  small  pins  s'  / ;  the  fnlcram 
«''of  this  lever  is  adjustable  by  the  two  small  screws  shenn  in  the  plan,F%i 
2.     A  long  notched  bar  k  is  fixed  to  the  under  side  of  the  table  m  by 
screws,  and  the   bar   t  has   its   fulcnim  f  also   fixed  to    it;    this   latter 
bar  t  is  moveable  on  its  centre,  and  may  be  placed  at  any  angle  by  an  sd- 
jnsting  screw  working  in  a  mortice  t'\  where  it  maybe  fixed  in  any  required 
position,  as  will  be  seen ;  it  is  by  the  angle  given  to  this  bar,  that  the  re- 
gular incrcatie  of  distance  is  obtained  between  the  rivets.    This  is  effected  in 
the  following  manner :  as  the  lever  e  works  on  its  centre,  the  pin  /  to 
which  it  is  fixed,  strikes  as  it  descends  on  the  end  of  the  lever/;  its  motion 
is  then  communicated  to  the  long  rod,  passing  under  the  table,  for  working 
the  lever  r  backwards  and  forwards,  it  being  fixed  to  the  rod ;  as  the  latter 
lever  moves,  it  alternately  engages  and  disengages  the  stop  or  projecting 
pieces  r  /  from  the  notched  bar,  allowing  it  at  the  same  time  to  slide  for- 
ward the  distance  between  the  rivets ;  while  the  bar  t  placed  at  the  required 
angle,  and  working  between  the  pins  /  s'  on  the  lever  «,  immediately  affects 
the  disuuice  tnivelled  by  the  lever  r,  which  it  shortens ;  by  this  arrangement 
the  machine  is  made  quite  self-acting,  but  it  may  also  be  worked  by  the 
handle  on  the  end  of  the  rod  J. 

When  it  is  required  to  change  the  punches  h  for  different  siied  holeSi  it  is 
necessary,  when  replaced,  to  adjust  them  to  the  greatest  possible  aeemacj  in 
the  dies  i ;  this  could  not  be  done  without  stopping  the  whole  line  of  diaft 


EXPLANATION    OP    THE    PLATES. 


4S5 


t,  were  it  not  for  a  provision  tiiaile  for  tLat  purpose ;  in  bucIi  a  caec,  the 

■in  the  upper  part  of  theeonoectitig  rod  c  is  withdrawn,  a  block  of  wood  is 

D  placed  between  the  ^me  and  the  lever  d,  whose  fulcrum  is  on  the 

e  of  the  frame  p,  upon  which  it  rests  ;  it  is   of  course   understood   that 

e  punching  lever  d  ia  at  rest,  being  disengaged  from  the  pin  e',  the  result  of 

is  the  working  of  the  long  mortice  of  the  connecting  rod  up  and  doHii 

Llie  pin,  without  communicating  any  motion  whatever ;  and  by  the  lover 

rorked  gently  by  hand,  and  shewn  dotted  in  Figs.  2  and  3,  the  puucliing 

rr  e  can  be  miscd  and  lowered,  communicating  its  alternate  motion  to 

slide  fmme,  by  which  means  the  punches  ore  with  great  facility  ad- 

Stcd  in  the  dies  f. 

3  jaws  in  the  frame  />  arc  much  strengthened  by  placing  a  square 
;ht  iron  bar  between  them,  in  places  provided  for  the  jiurpose. 
im  llie  above  description,  it  uill  be  seen,  tlint  the  advantages  pos- 
i«l  by  this  machine,  more  than  compensate  for  the  increased  number 
3  parts,  which  are  but  few  when  compared  with  its  superiority  over 
e  of  ordinary  construction  ;  there  are  several  at  work  in  the  boiler  shop 
r  Majesty's  Dock  Yard,  Woolwich, 

PLATE  LII.  A. 

Steam  Punching  Maciiinb,  by  M.  Cave,  Paris. 

The  mode  of  applying  the  motive  power  to  this  machine  is  altogether  on 
a  different  principle  from  the  others  contwned  in  this  work,  to  which  it  is 
either  conveyed  through  wheels,  pulleys,  straps,  or  bands,  driven  from  the 
shafting  running  through  the  building.  In  this  case  it  is  worked  by  a  small 
steam-engine,  connected  to  and  forming  part  of  the  machine.  In  the  steam 
^linder  a  is  a  solid  piston  and  piston  rod  b,  accurately  fitted,  the  cylinder 
Lbeing  bored  out  in  the  usual  way ;  (^  is  a  shde  valve,  worked  bv  the  rod  d  on 
e  faces  of  the  ports  c'  and  c",  the  former  for  tie  admission  of  steam  to  the 
f  ojrlinder,  and  the  latter  for  the  exhaustion,  whence  it  is  carried  through  a 
pipe  to  any  convenient  outlet ;  the  rod  d  works  steam  tight  throui;h  the 
etuffing  box  of  the  slide  case,  which  contains  the  steam  brought  by  the 
a  pipe  e  from  the  boiler.  This  latter  pipe  is  connected  to  the  valve 
iwng,  by  flanches  bolted  together. 
[  On  the  lop  of  the  piston  rod  A  is  a  cross  head  6',  on  either  side  of  which 
e  the  links  yy  connected  to  the  punching  lever y";  two  small  rollers  &' b'. 
Fig.  3,  are  placed  on  the  outer  ends  of  the  cross  head  which  slide  up  and 
down  in  the  guides ,/""/"",  whereby  the  parallel  motion  of  the  piston  is  kept 

I  perfectly  true.     On  one  end  of  the  punching  lever  _/  is  the  connecting  rod 
t,  for  conveying  the  altoriiatc  or  reciprocating  motion  of  the  piston  tltrough 


I 


^Lbeing 
»liefi 
^Fojrlint 


4>G6  EXPLANATION    OF   THE    PLATES. 

the  crank  /  to  the  fly  wheels  ira,  by  which  it  is  r^ahted.  Oil  die 
end  of  the  lever  f  (whose  fulcrum  is  on  the  frame  A)  is  the 
cylinder  for  holding  the  punch,  both  of  which  have  projeetiiig  pins  ob 
for  connecting  them  together  by  the  wrought  iron  links  ^  ^^  for 
the  parallelism  of  the  punch,  these  links  being  adjostable  by  the 
above.  The  punch  n  is  connected  to  the  lower  end  of  the  cjlinder  $\^% 
key,  as  shewn  by  Fig.  8,  while  the  die  o  in  which  it  works  can  be  ahcnii 
according  to  the  size  of  the  punch  used,  by  the  two  small  adjoBtiiig  tovn 
shei^-n  in  Fig.  1 .  A  stop  p  prevents  the  plate  from  rising  after  it  has  bees 
punched,  the  circular  pieces  punched  out  falling  through  the  apertme  «  pn^ 
vided  for  that  purpose. 

The  strong  frame  h  for  canying  the  several  parts  of  the  machine,  is  »- 
curcly  fixed  to  the  stone  foundation ;  on  the  front  part  of  this  finame  is  s 
cap  (fixed  by  six  bolts  and  nuts,  Fig.  4,  representing  a  sectional  plan  of  tlw 
frame)  which  is  tightened  according  to  the  wear  of  the  cylinder  g^  ^riuch 
becomes  considerable  after  having  been  at  work  any  length  of  time  ;  by  tUi 
means,  any  irregularity  in  the  motion  is  entirely  obviated. 

The  long  lever^  is  fixed  to  the  rod  d  for  alternately  opening  and  ■l»w**iiig 
the  ports  (f  and  c'^  by  the  slide  valve,  which  operation  is  performed  by  Ae 
lever y^  as  it  rises  and  falls,  striking  against  the  pins  cTaT,  the  counteibafaneB 
equalizing  the  weight  of  the  opposite  side  of  the  lever ;  whose  falerom  is 
screwed  to  the  upright  guide  frames  f'\  and  projects  consideiably  ow 
them,  Fig.  1 .  The  handle  /'  is  connected  to  the  lever  y,  for  starting  or 
stopping  tlie  engine  by  hand ;  and  by  means  of  the  pinsy'  on  the  two  up- 
rights fixed  to  the  upper  side  of  the  frame  h^  the  lever  j  is  secured  when 
the  machine  is  at  rest. 

In  its  present  position  the  working  of  this  machine  is  as  follows :  steam 
being  admitted  by  the  pipe  e  to  the  slide  casing,  passes  through  the  stesm  port 
</  to  the  under  side  of  the  piston,  which  it  presses  up,  causing  the  levery*  to 
rise  between  the  guides /*"  /*",  and  assume  the  position  represented  by  the 
dotted  lines,  the  connecting  rod,  crank,  and  consequently  fly  wheel  following 
their  respective  motions.  In  this  new  position,  the  levery  moving  on  its  ful- 
crum, must  necessarily  depress  the  punching  cylinder  g  and  punch  w,  thereby 
communicating  to  it  the  requisite  alternate  motion.  In  its  ascent,  the  lever 
y* strikes  against  the  upper  pin  d\  and  raises  the  valve  over  the  steam  port 
r,  which  is  no  sooner  done  tlian  the  steam  that  has  performed  its  duty 
rushes  through  the  exhaust  port  (the  communication  being  made  between 
the  two)  and  through  it  makes  its  esca})e,  when  the  piston,  rod,  lever,  &e^ 
fall  by  their  own  gravity  to  tlieir  original  position,  (having  merely  to  over- 
come the  pressure  of  the  atmosphere,)  till  the  lever  on  its  descent  again 
strikes  on  the  lower  pin,  which  inunediately  opens  the  steam  port  ^  to  die 


EXPLANATION    OF   THE    PLATES.  -tdj 

posidon  shewn  by  Pig.  i,  which  has  already  been  described, — when  tho 
operBtion  is  agaia  repented. 

This  machiDC  may  be  used  for  catting  the  edges  of  plates,  byadoptiug  the 
cutter  ahewu  by  Figs.  5,  G,  and  7- 

AU  the  parts  of  an  liigh  pressure  steam-engine  ore  necessary,  as  already 
described,  to  work  this  description  of  punching  machine,  oud  a  higher  ve- 

Ihwdty  to  the  punch  must  necessarily  be  given  than  by  the  ordinary  metiiod. 
I  Tt 


BivBTiNo  A 


PLATE  LIII. 


INK,  BV  Messrs.  Paihbacsns  and  Co. 


The  drawings  shewn  by  Figs.  1  and  2  very  much  reeemhle  the  com- 
in  punching  machine,  the  motion  being  in  every  respect  similar.  The 
work  performed  by  this  machine  is  in  all  ordinary  cases  done  by  ma- 
nual labour,  which  operation  necessarily  requires  the  services  of  three  men, 
the  bolder  on,  to  hold  his  hammer  or  tool  inside  the  boiler  against  thu 
of  the  rivet,  while  tlie  other  two  beat  out  the  iron  to  the  conical  form 
to  its  opposite  end;  this  is  attended  with  a  very  disagreeable  noiac, 
[nite  unoToidable,  but  which  is  entirely  done  away  with  by  the  nse  of  lliia 
machine,  which  performs  with  almost  instantaneous  pressure,  what  is  done  in 
all  ordinary  cases  by  a  long  series  of  impacts.  Fig.  1  is  an  elevation,  and 
3  a  plan,  where  it  will  be  seen  that  the  motion  is  commuuicated  to  the 
[ht  and  loose  riggers  a  a  by  a  leather  strap ;  the  speed  at  which  these  re- 
is  much  too  great  for  the  purpose  intended  by  tlus  machine,  consc- 
.tly  a  pinion  and  large  spnr  wheel  b  is  required,  by  which  it  is  reduced  as 
istol,  thatis  while  the  pinion  shaft  is  making  six  revolutions,  that  on  which 
luge  wheel  is  hung,  and  also  the  cam,  is  revolving  at  only  one.  On  the 
pinion  and  rigger  shaft  is  placed  the  fly  wheel  c,  for  giving  a  unifunn  motion 
to  the  working  parts  of  the  machine.  The  riveting  lever  e  is  then  mode 
to  rise  and  fat!  hy  the  action  of  the  cum  d,  the  face  of  which  is  steeled,  wliilo 
on  the  end  of  the  lever  e  is  the  steel  roller  /,  revolving  as  the  cam  works 
gainst  it;  hy  this  contrivance  the  friction,  which  would  bo  considerable,  is 
materially  obviated ;  the  riveting  lever  works  easily  on  its  fulcrum  A,  the 
two  short  links  connecting  it  to  the  riveting  too!  j,  sliding  backwards  and 
forwards  in  the  socket  bearing,  by  which  it  is  kept  working  in  a  perfectly 
troe  and  horizontal  direction. 

The  various  parts  of  this  machine  are  connected  to  the  side  frame  •;, 
made  of  cast  iron,  and  on  it  are  placed  the  plmmncr  blocks  for  carrying  iJie 


I  On  the  sole  plate  is  fixed  the  riveting  block  i,  agmnst  which  Uial  pait 


4G8  EXPLANATION  OF  THE  PLATES. 

the  boiler,  chimney,  or  other  work  k  being  rivets  is  placed ;  it  is  Im^ 
by  the  block  and  chain  shewn  in  Fig.  1 . 

The  operation  is  thus  performed :  the  rivet  being  put  through  the  holeslif 
the  attendant  workman,  previously  punched  by  the  punching  tw^Ki^ft^  it 
brought  round  so  that  its  head  will  fall  into  the  recess  on  the  projecting 
part  of  the  riveting  block ;  the  machine  is  then  put  in  motion  hy  rhenging 
the  position  of  the  strap  from  the  loose  to  the  fixed  pulley,  which  bsii^ 
communicated  to  the  riveting  tooiy,  gives  the  required  shape  to  the 

The  whole  is  placed  on  a  suitable  foundation  /  for  giving  solidity  and 
bility  to  the  machine,  and  a  pit  is  made  to  allow  the  wheels  to  wotIl  in. 

It  is  stated  by  Messrs.  Fairbaim,  that  with  the  attendance  of  two 
and  two  boys  to  the  plates  and  rivets,  this  machine  can  ^tl  in  the  finnest 
manner  eight  rivets  of  three  quarters  of  an  inch  diameter  in  a  minute^ 
whereas  by  the  common  process,  three  men  and  a  boy  can  only  rivet  op  40 
per  hour,  the  qiumtity  then  done  in  the  two  cases  being  in  the  piopoitian  of 
480  to  40,  or  as  12  is  to  1,  exclusive  of  the  saving  of  one  man's  lahonr. 

PLATE  LIV. 
Double  Gkindino  Machine,  by  Messrs.  Nasmyth,  Gaskell,  and  Oo. 

The  object  of  the  machine  represented  by  this  Plate,  is  to  grind  up  the 
f\u;e8  of  the  different  parts  of  machinery,  when  a  great  surface  is  reqnired 
to  be  made  ]>erfoctly  smooth ;  to  accomplish  which,  this  contrivance  has 
been  used. 

Figs.  I,  2,  and  3,  severally  shew  a  side  elevation,  an  end  elevation,  and 
a  plan  of  a  double  face  grinding  machine,  the  one  side  being  a  repetition  of 
the  other.  To  the  two  cast  iron  cross  frames  a  a,  are  bolted  two  large 
plunimer  blocks  for  carrying  the  main  shaft,  having  at  each  extremity  the 
circular  frames  divided  into  twelve  compartments,  in  which  are  placed  the 
grinding  stones  f^  each  being  adjustable  by  the  small  set  screws  m  round  its 
circumference.  On  the  top  of  the  cross  frames  a,  are  placed  two  longitn- 
dinal  frames  b  h^  made  also  of  cast  iron,  for  supporting  the  long  bed  frames  c  c, 
and  also  the  self-acting  apparatus  furnished  to  this  machine.  Two  motions, 
the  one  at  right  angles  to  the  other,  are  given  by  the  slides  d  working  along 
the  beds  r,  and  also  the  face  plates  e  for  carrxnng  the  work,  by  which  it  is 
brought  into  contact  ^ith  the  grinding  stones.  Pits  are  made  to  allow  the 
wheels,  ior  carrying  the  stones,  to  work  in. 

Tlie  self-acting  motion  given  to  the  work  being  faced,  by  means  of  which 
it  slides  along  the  bed  c  while  the  grinding  stones  are  revolving  on  their 
axes  i-^  thus  obtained :  on  the  main  shail  next  to  the  driving  riggers  or  pul- 
leys ^,  is  a  worm  /,  which,  as  it  revolves,  works  a  worm  wheel  rejnresented 


EXPLANATION  OF  THE  PLATES.  469 

by  the  dotted  lines  in  Fig.  2,  tbos  commimicatmg  ihe  motion  to  the  upright 
spindle ;  from  this  it  is  carried  by  the  bevel  wheels  to  the  spindle  running 
horizontally  the  whole  length  of  the  machine,  having  at  each  extremity  three 
small  bevel  wheels.  The  action  of  this  apparatus  is  thus,  supposing  the  slide 
to  be  travelling  in  the  direction  towards  the  small  bevel  wheels,  two  of 
which  are  required  for  the  purpose,  while  the  third  or  outer  one  runs  freely 
on  the  spindle,  without  producing  any  effect,  the  small  clutch  being  dis- 
engaged from  it ;  on  the  travelling  slide  d  is  fixed  a  stud  or  pin  A' ;  a  long 
rod  h  of  the  same  length  as  the  bed  e,  is  moveable  in  two  stud  bearings 
fixed  to  it  As  the  slide  d  travels,  the  pin  A'  comes  into'  contact  with 
a  second  stud  or  pin  adjusted  to  any  position  on  the  rod  h^  according  to 
the  length  of  the  motion  required,  which  must  naturally  press  it  forward, 
and  thereby  throw  out  the  clutch  on  the  end  of  the  spindle,  which  being 
shifted  from  one  bevel  wheel  to  the  other,  disengages  that  which  had  been 
at  work  before,  while  it  engages  the  outer  one,  that  had  been  running 
loosely  on  the  spindle ;  by  this  curious  contrivance,  the  screw  for  working 
the  slide  revolves  in  a  contrary  direction,  and  instead  of  drawing  the  slide 
d  towards  it,  sends  it  back.  A  counterbalance  weight  n  is  connected  to 
the  extremity  of  the  rod  h^  for  keeping  it  in  a  steady  position  while  this 
operation  is  being  performed. 

The  tappet  wheel  k  fixed  on  the  end  of  the  screw  for  advancing  the 
other  slide  e,  is  also  worked  by  a  pin  on  the  same  rod  A,  whereby  the 
work  is  advanced  to  the  face  of  the  grinding  stone ;  it  is  on  the  upper  part 
of  the  slide  e  that  the  work  is  fixed.  A  substantial  foundation,  consisting 
of  stone  work,  is  prepared  for  receiving  the  two  frames  a  a,  and  to  which 
they  are  firmly  bolted  down  by  strong  holding  bolts. 

Another  mode  of  performing  this  same  operation  might  be  adopted,  by 
fastening  a  whole  grindstone  into  the  chucks,  and  passing  a  bolt  through  two 
surface  plates  of  two  feet  diameter  each,  one  on  the  middle  part  of  each 
face  of  the  grindstone,  by  which  means  they  would  be  more  effectually  se- 
cured in  their  places. 


f 
I 


f 


r. 


i 


,♦ 


*i 


^    r 
-<    I 


»"! 


ii 

• » 

;  ■ 


INDEX. 


Action,  manner  of  .        .192 

Alder,  strength  of  .  .  250,  251 
Alloys,  strength  of  .         •     260 

Alteration  of  velocity  by  friction  836 
Alternate  cones  .  .  .  531 
Anderson  ....  485 
Angle  defined  ...  71 
Arc  of  a  circle  defined  .  .128 
Arkwright,  invention  of  cotton 


machinery  . 
Ash,  strength  of 
Asp,  strength  of 
Axis,  defined 
Axles,  hollow 
■  solid    . 


252.  394.  428 

.      251. 253 

.     253 

.     410 

.      231. 236 

.     353. 369 


Banks,  John  . 

Barlow,  Peter,  experiments 

Bayonet 

Beams   .        .        •        . 

Bearings  of  shafts  . 

Beech,  used  for  patterns  . 

— — —  pillows    . 


strength  of  . 


Bending 
Bevel  gear 
—  wheels . 
Birch,  strength  of    . 
Bismuth,  strength  of 
Blocks   . 
Bodies  of  shafts 
Bone,  strength  of    . 


.  351 

by  259 

.  295 
338. 352 

.  456 

.  128 

.  346 

.  251 

.  222 

.  51 
51.62 

.  253 

.  260 

.  344 

.  221 

.  252 


Page 
.      270 

.  273 
.  253 
.  346 
.  347 
253. 261 
258.  260 
.  344 


Boring  mill  clutch  . 
Boulton  and  WaU  . 
Box-wood,  strength  of    . 

used  for  pillows 

Bramah,  Joseph 
Brass,  composition  of 
—strength  of   . 
Brasses  .... 
Breadth  of  teeth  of  wheels 

95.  97.  200 

Breasts 347 

Breast  wheels,  power  of  .        .    332 
Brewster,  Dr.  ...      64 

Brick,  strength  of   .         .        •     254 
Britain,  manufactures  established  in 

172.  177 
Brown,  B.,  experiments  by  255, 256  ' 
Bucket  water-wheels,  weight  of  204 
Buffon,  experiments  by  .  •  255 
Bums,  Robert  .  •  •  236 
Bushes 347 

Camus,  on  the  teeth  of  wheels 

6. 10.  80.  65 


Capstan  bar    . 
Carmichael,  James  • 
Cast  iron,  introduction  of 

bearings  . 

■  gudgeons 

framing  . 

■  pinions   . 


•    285 

.    107 

.     274 

.    344 

198.  205 

350.854 

50.56 


shafts    287.240.850.858 


472 


INDEX. 


CSast  iron  staves 

■  strength  of 
— ^-^—  trundles  . 


Page 
23.  113.204 

.     251 

.       35 

274. 422 

.     251 

2 


Cattle  mills    . 

Cedar,  strength  of  . 

Centres,  line  of 

Centre  of  gravity  of  shafts  loaded 

with  2,  3,  or  4  wheels       354.  372 
Charts,  construction  and  use  of 

129.  131 
Cherry  tree,  strength  of  .  .  253 
Chord  of  an  arc  defined  .  .  73 
Circle  defined  ...       72 

Circular  arcs  for  describing  teeth 

of  wheels    ....       23 

,  Willis's,  for  the  same 

148.  151 

Cloves 349 

Clutches  and  glands  .         .     269 

Clutch,  boring  mill  .         .         .     270 

second  construction      .     271 

Cogs,  term  explained  .  .  1 
Cohesive  strength  of  bodies  250.  262 
Conductor  ....  37 
Cones,  alternate       .         .         .     336 

defined         ...       75 

friction         .         .         .     302 

proportional  .         .       53 

Connection  of  shafts  .  .267 
Copper,  strength  of  .         .250 

alloys  of     .         .         .     260 

Com  mills,  numbers  for  .         .     215 

tackle  for       .         .     299 

Corollary  defined  ...  73 
Cotton  mills  .  .  .  .264 
Coulomb's  experiments  on  friction 


Coupling  box  . 

link 

Couplings,  with  one  bearing 
— ^  with  two  bearings 


306 
267 
273 
276 
267 


Couplings  for  aprigbt  d 

with  gknds 

— — —  durability  of 
— — — •  round 

■  self-disengagiiig 

square    .         .      267. 

Crab  tree,  strength  of 
Crompton,  Samuel  . 
Cross-tailed  gudgeons 
Cubical  parabola 
Curves,  epicycloidal 
Cycloid,  form  of  teeth 
Cylinder  cutters  of  a  defined 
diameter  .  .  .  75. 
Cylinder,  solid,  strength  of 

hollow,  strength  of  229. 

Cylindrical  shafts     . 


27S 


a04 
277 


278 
17i 


54 

155 
197 
241 


iron    . 
Cypress,  strength  of 


-  hollow,  of 


240 
251 


Dash  wheels  ....  280 
Deal,  strength  of  .  .  ,  253 
Decay  of  timber  .  .  .  349 
Desaguliers,  Dr.  ...  88 
Diameter  of  pitch  line  •  •  55 
Disengaging  machinery  .  .291 
Donkin's  table  of  radii  of  wheels  114 
Double  speed,  method  of  obtain- 
ing     426 

.  278 
.  97 
.  306 
.  290 
79.83 


Drum  shafts   . 
Du'Buat 
Duncan,  John 
Durability  of  couplings 
■  wheels 


Eclectic  Review,  cxtiuct  from 

173, 174 
Edge  stones  .  •  •  .  337 
Egg-formed  pivots  .  •  .  847 
Elder,  strength  of  .        •        .    258 


INDEX. 


473 


Page 

Elm,  streDgth  of     .         .      253. 257 
Emerson,  W.  .         .      253. 351 

Epicycloids,  properties  of     •       ?•  17 

mode  of  describing  8. 16 

.  8.13 

.       12 

54.  57.  59 

.       11 

16.17 

16 

17 

175 

176 


exterior 

interior 

spherical 

Exterior  epicycloid  • 
Epicycloidal  curves 

lengths  of 

■ areas  of 

Essay  on  the  shafts  of  mills 
■  how  treated 


Face  wheel  .  .  .  .336 
Fast  and  loose  pulley  .  .  297 
Feathered  shafts  .  •  .181 
Feathering  ....  352 
FeeUng,  a  term  .  .  .  222 
Fen  ton,  Murray  .  •  .10 
Fen  wick,  Thomas  .  .  .191 
Figure,  best,  for  teeth  of  wheels  14 
Figures  iUustrating  the  slide  rest 

398.  405 


Fir,  strength  of 

shafts 

Flanks  of  teeth 
Flexure 
Flour  mills 
Fly-wheels 

shafts 

Force  defined 


immediate, 


and  horses  . 
Fracture 
Framing  of  mill-work 

'  for  lying  shafts 

— —  upright  shafts 

for  flour  mills 

cast  iron    • 

wooden     • 


251.  253 
.  243 
.  37 
.     221 

284. 353 
.  212 
.     216 

.    .77 
of  men 

.  89 
336.351 
342.  344 
.  348 
.  349 
.  353 
.  350 
.  348 


Free-stone,  strength  of    . 
Friction  of  teeth  of  wheels 


cones     . 
clutches 


loss  of  power  by 

rollers 

wheels  acting  by 


.  254 

29.67 

.  302 

.  301 

.  329 

.  346 

.  303 


Galileo 228 

Gear 51 

spur       .        .        •         .18 

bevel      .         .         .        .51 

Generating  circle     •         .        .  7. 50 

Glands 269 

Gold,  strength  of  .  •  257, 258 
Governors,  principles  of  •  .  312 
steam  engine  .        .    308 

water-wheel   .        .313 

first  construction      .     313 

second  construction      315 

third  construction    .     315 

fourth  construction  .     316 

fifUi  construction     .     317 

windmill         .         .     311 

Gravity,  centre  of,  rules  for  find- 
ing that  of  shafts  loaded  with 

2,  3,  or  4  wheels  .  3.54.  372 
Gregory,  Dr.  Olintfaus  .  .  232 
Grindstones,  spindles  of  .  .  210 
Gudgeons,  mode  of  fixing        .     200 

diameters  of  .        .     197 

cast  iron         .         .     205 

malleable  iron         .    207 

strength  of  .     198 

viratei^wheel   .      199,  200 

stress  and  friction  of    184 

pressure,  greatest,  of    194 

table  of  .        .        •     202 

of   cast    and 

wrought  iron  •  .  .  210 
Gun  metal      ....    860 


474 


INDEX. 


Halley,  Dr.,  on  qiicjcloids 
Hatton  on  clock  work 
Haxel,  strength  of  . 
Headstock  framing 
Heating,  to  prevent 
Hempen  rope,  strength  of 
Hewes,  of  Manchester 
Hollow  axes  . 

cylinders     . 

shafts 
Holly,  strength  of  . 
—  used  for  pillows 
Hook,  Dr. 
Horn,  strength  of    . 
Horse  mill,  or  oxen 

power  . 
^— — ^—  performance 

men  and  horses   • 
Hatton,  Dr.  Charles 


Imison   . 

Inequalities  of  teeth 
Inertia    . 

Interior  epicycloid  . 
Intermediate  spindles 
Internal  pinion 
Involute  teeth 

properties  of 

length  of   . 


Page 

17 
125 
253 
848, 849 
345 
253 
317 
231 
241 
181.240 
253 
846 
277 
252 
190 
88 
of,  hy 

90 
372 


27 
28 
292 
8 
206 
47 
44.  64 
63 
44 


Iron  bar,  strength  of      251.  255,  256 
cast,  strength  of         .257 

■  for   strength  of  mate- 
rials  ...  .         83. 88 

■  demonstration  of  power 

of 254 

sliafts — see  Shafts. 

Ivory,  strength  of    .         .         .252 

Jacks,  form  of  teeth  for  .         .       48 
Jamiesou's,  Dr.,   Mechanics  for 
Practical  Men     .         .        77.359 


Jennies,  common 
Johnson,  Dr.  • 
Joints,  aniversal 
Journals 

—  table  of 

cast  iron 

— ^  of  shafts, 


Joumejrs  or  Joomals 
Jujeb,  strength  of   • 

Kelly,  William 

Kyan's  preserving  timber 

La  Hire 
Lantern . 
Lateral  stifihess 

strength 

stress 
Lead,  strength  of    . 

allo3r8  of 

Leaves,  defined 
Lemon,  strength  of 
Lift 

tenters     . 

— — —  for  windmills 

Lignum  vitffi  . 

Line  of  centres 

Lock  pulley    . 

Locust  tree,  strength  of 

L}ing  shafts     . 


278 
48 

m 

SIS 

su 
sia 

SIS 

S7 

857 


.    MS 

.      71 
1 


cast  iron 
malleable  iron 


251.  S5S 

•  S0O 

.     l.« 
.     S51 

.  868 

.  805 

.  311 

•  diff 

8 

.     897 

.     851 

849.348 

.     839 

'.     841 


Machines,  power  of,  men  and 

horses         •        •        •  •      90 

Machinery,  remarks  on    .  .178 

changing  velocity  of    334 

Malleable  iron  shafts        .  .     849 

strength  of  .     850 

Materials  for  patterns      .  .188 

Maximum  effect  90»  91 


^5 


pechanics  for  Practical  Men 
Bchaniciil  power  explttined 


78 


-  substitute    for    the 


79.83 
311.347 


S42.  344 


humELn  hand 
Men,  strength  of 
Metals,  strength  of . 
Meux  and  Co. 
Mill-wheel  work,  shafts 
■  •-*—  stones,  on  6xing 
^mf^ —  Telocity  of 
^^^•^  work,  on  fruntog 
^fkotnentuin  defined 
^Hlorvcau,  Gayton  .     257,  258 

^piotioD,  of  the  methods  nsed, 
when  motion  is  conveyed  to 
wheel-work         .        .         .     299 

mechanism    for    eqiiaJ- 

iztng  the  motion  of  mills      ■     305 

— uniform      .         .         .323 

Holherry  tree,  strength  of        .     251 


chan^g  velocity  of 
ly,  rule  for  teeth 


.     278 
.     337 


ihro^k's  experiments 


myth's  remarks  on  the  intro- 
'  dnction  of  the  sUding  principle 
in  tools  and  machines  employed 
in  the  production  of  machi- 
nery   393 

Newton,  Sir  Isaac  ...  76 
Nicholson,  Peter  ...  332 
Nicholson's  journal  .         .     349 

Numbers,  for  arranging,  for  wheel 
work  115.117 

-  for  horse  engines 


Oak  shafts  .  .841 

Odontograph,  Willis's  .  .  168 
Oil,  effects  on  couplings  .  .  286 
Overshot  wheels,  experiments  on  232 

,  theory  of       .     326 

^~^—^— ,  power  of       .     330 


■f  velocity  of 
-,  weight  of 


328 


Parabola 
Parallel  lines  . 

motion 

Parent,  experiments  by 

Parry,  Dr. 

Pasteboard  used  for 

Pattern  teeth  . 

Patterns,  on  making 

Pedestals 

Peel,  Williams,  and 

Pendulums 

Perpendicular 

Pillow  block  . 

Pillows  . 


beatings 


friction  of    . 

internal 

to  find  the  fignre 

wheel,  &C. 

Pitch     . 
table  of 

Pitch  pine,  strength  of 
Pi  tot,  experiments  by 
Pivots,  fonns  of 

pressure  on  . 

Phine  tree,  strength  of 
Planing  machine 
Platinum,  strength  of 
Plumber  block 
Plum  tree,  strength  of 


27 
.  134,  125 


46,47 
of  the 

21,25 


257 
347 


-176 


Polygon  defined       .         .         . 

73 

Round  conpling 

M.n 

Pomegtnnatc  tree,  etrengtb  of  . 

251 

Romford.  Const       . 

.  w 

Poplar,  strength  of  . 

251 

RDpp,ofH»cfaMter 

.     M 

Power,  nature  of    .         .          78,  77 

. mechanical  . 

78 

Sack  tackle     . 

.    M 

hortea' 

88 

ScrewB  .... 

.    M 

of  water  wheel      . 

78 

SemicaUd  puobok         . 

.  m 

330 

Shafts    .... 

.  m 

PrimitiTe  ndii 

3 

geometrical  figurea     , 

39* 

how  made  till  of  Ikte 

.  m 

Principal  diameter  . 

55 

framing  for  npri^t 

.    SM 

Proportions  of  melalg 

261 

on  the  bodies  of 

.  HI 

3 

53 
S8 

lateial  atifibess  and  html 

n^    .        .        . 

2 

table  of. 

.    241 

75 

proportion  of . 

.    211 

Pump  mochinery,  numhcre  of  . 

123 

wrought  iron,  to  ttn 

Mb- 

Pnlley,  fast  and  loose 

297 

teral  streas  of      . 

.    Ml 

lock     .... 

297 

cast  iron 

181.  ssr 

eliding          .         .         . 

294 

cyUndrical              17». 

S88.I44 

hoUow  . 

.    MO 

Quince  tree,  strength  of. 

251 

square   .                223,  224,  2X5 

stress  upon     . 

.     183 

Back  and  pinion,  tcetli  for 

48 

to  rcwst  torsion 

232.235 

Radius,  to  find  tlic,  of  pinion    . 

50 

ivooden         .      179, 

180.  242 

Radii  of  wheels,  table  of  .     114 

,115 

Sickengcn 

.     257 

real        .         .         .     3.  24..  40 

-Sliver,  strength  of  . 

251. 257 

Red  fir,  strength  of. 

253 

Slide-rest  principlo , 

.     898 

Re-eiignging  machinery    . 

300 

Sliding  pulley 

.     294 

Ronnie,  George                      256 

,257 

.     192 

RciKJrtory,  No.  73  . 

349 

Smcaton,  John 

.     283 

Revolving  pendulum 

309 

Solid  axles     . 

353.  369 

Robcrton,  John,  on  teeth  of  wheels 

Soufflet  .... 

.     255 

103 

Southern 

.     210 

no  rlinft'' 

183 

Spindles 

Spmdle  grindstone  . 

i,,r...  .r„.u.,~u 

.    847 
.     «0 

Robison,  Professor  .        64.  233 

269 

Rondelct,  experiments  by 

857 

Spurgear       .        .         . 

18,18 

Rothsay  cotton  mill,  experiments 

wheeh,  .         .         . 

.       62 

on 

318 

-    877 

INDEX. 


477 


Square  shafts. 
Squeezers 
Staves    . 

cast  iron 

strength  of  . 

indefinitely  small 

when  to  be  used 

Staye-formed  teeth . 
Steam  engines 

governors 

Steel  pivots    . 
strength  of 


Page 

267 

353 

J.  18 

37 

113 

19.  21 

37 

19 

176 

308 

.     347 

,     308 

.     346 

.     226 


Steps,  form  of 

Stiffness 

Strain,  measured  by  horses'  power  87 

Strength  .         .         .         .226 

of  gudgeons  .  .198 
of  horses    .         .         .113 

■  of  iron        .         .         .     250 

■  of  men  .  .  89. 94 
of  shafts  .  .182 
of  staves     .         .         .113 

'  of  teeth      ...       53 

■  of  timber    .         .         .     253 

Stress,  lateral ....     228 

on  shafts  and  gudgeons 

179.  197.  212 
on  teeth  of  wheels        86. 108 

Tables  of  gudgeons 

202.  205,  206.  210 

of  journals   .         .         .     216 

of  pitches  of  wheels       .     114 

— ^—  of  shafts  of  cast  iron  240,  241 

solid        .      240, 241 

hollow  240,  241 

—  of  squares,  cubes,  &c.         242 

of  wheels     ...       95 

how  to  be  con- 
sidered, &c.         .         .         .175 
Tamarind  tree,  strength  of  257 


Tangent  of  a  circle 
Teak  wood,  strength  of  . 
Teeth,  principles  of 
method  of  forming 


Page 

73 
254 

4 
5 


described  by  circular  arcs      23 
of  bevelled  wheels  55.  62 


95.97 

18.  22.  25 

5.  8.  16 

66.70 

.       28 


breadth  of    . 

epicycloidal  .  * 

form  of 

friction  of     . 

inequalities  of 

involute 

to  describe  the  teeth  of  a 

wheel  for  a  trundle,  by  means 

of  a  circular  arc  . 

indefinitely  small,  as  of 


50 


23 


one  cylinder  rolling   on   an- 
other .... 
—  to  determine  the  breadth 


of 


of  wooden    .         .         . 

figure  of,  when  the  staves 

of  the  trundle  are  cylinders  of 

a  finite  diameter  . 

— ^-  length  of 

strength    of,  in 


111 
112 


power 

thickness  of 

wear  of 

wooden 


21 

44 


horses' 

83. 106 

109,110 

.       29 

29.112 

.     255 

.     278 

.     308 

.     251 

.     260 

.     260 


Telford,  Thomas     . 
Throstle 
Throttle  valve 
Timber,  strength  of 
Tin,  strength  of 

alloys  of  . 

Tools,  the  making  of  and  tem- 
pering        ....     410 
Torsion,  on  shafts  subject  to 

231.  243 
Tremor,  effect  of    .        .     343,344 

1 1 


I 


I 

I'l 

i 


t 

\ 


t 

9 

I. 


I 

I    I 

I    , 

I' 

i 


■•  ■!■■:■ 


r 


INDEX. 


*79 


Wooden  abafts 
— — -  teeth,  breadth  of 
'      thickness  of 

wheels,  weight  of 

Wrought  ironr—see  Iron, 

Yew,  strength  of    . 


242 
112 
110 

204 


258 


Young,  Dr.  T.        •  5. 65 

•—  letter  from,  to  Bucfa&mui 

66.71 


Zinc,  strength  of     . 
—  Indian  . 
Goslar   • 


.  260 
.  260 
.    260 


THB  BND. 


O.  WoodftU  SDd  Son*  Prtattm^  Aflgil  Ooort,  SklHMT  Sliittt 


•1 


\ 


f  1' 


'•I  r 

r 


I 


i 


■■1^-'-.v 


>-t-;l.v     ■>     X'